The Ideal Gas Law Practice Problems

A bio sauna operates at a temperature of 55˚C with an average relative humidity of 65%. The vapor pressure of water at 55.0°C is 1.78 × 10⁴ Pa. Calculate the partial pressure of water vapor inside the sauna. Relative humidity is the ratio of water vapor partial pressure in air to the vapor pressure of water at the same temperature.

Fixed-volume containers can be used as thermometers. The temperature of a sample is reported in terms of absolute pressure. If such a thermometer displays an absolute pressure of  420 mm Hg at the triple point of carbon dioxide (CO₂), determine the pressure displayed at the boiling point of ethanol.

The triple point for carbon dioxide occurs at -57°C and 5.2 atm. The pressure of a gas sample at this triple point is 3.8 atm. If the volume does not change, determine the pressure of the gas sample at the boiling point of nitrogen.

A sealed light bulb is filled with argon molecules at normal atmospheric pressure of 1 atm and a temperature of 293 K. What is the average spacing between two argon molecules?

During an experiment, a researcher measures the pressure of nitrogen gas and the temperature inside the cryostat chamber and finds that the pressure is 10^-7 mbar and the temperature is 50 K. The volume of the cryostat chamber is 10 cm³. How many nitrogen molecules (N) are present inside the cryostat chamber?

The mass of a molecule of substance K is 1.297 ×10^-22 g. Calculate the molar mass (M) of substance K.

During a chemical vapor deposition experiment, a cylinder containing 10 kg of diatomic nitrogen (N₂) is used. Calculate the number of i) N₂ moles (n) and ii) N₂ molecules inside the cylinder. The molar mass of nitrogen is 28.0 g/mol.

What is the radius (R) of a sphere containing 200 billion helium gas molecules at 300 K and at a pressure of 10⁵ Pa?

The pressure of the air inside an insulation system of volume 10 cm³ is 10^-8 atm. The air temperature is 250 K. The pressure rises to 10^-4 atm due to a system malfunction. Compare the number of molecules that would be present at the same temperature inside the system before and after the system stopped working properly.

A container with a movable piston is filled with SO₂ of molar mass 64.07 g/mol. The movable piston keeps the pressure inside the container at a constant 924 Pa. The container is moved back and forth between  100°C and -10°C. Determine the ranges of the density (in mol/m³) of the SO². 

The service ceiling is the maximum operating altitude of an aircraft. A military helicopter has a service ceiling of 8,000 m, where the air temperature is -36.9°C, and the density equals 0.53 kg/m^3. At that altitude, what is the atmospheric pressure (p)?

A balloon made of rubber and filled with air has a volume of 1 m³ under standard conditions of pressure and temperature. What is the volume of the balloon (V_Mars) if it is placed outside a Martian station where the pressure and temperature are 600 Pa and -60 °C, respectively? Assume that the balloon will not burst on Mars.

A laboratory technician entraps 2 ml of air at a pressure of 0.300 atm inside the barrel of a syringe with a tightly closed hub during an experiment. He increases the volume of the air to 6 ml by gradually moving the plunger back. The temperature of the air inside the barrel decreases from 25°C to 20°C. What is the pressure's final value (p_f)?

Find the density of the air (ρ) inside a truck tire filled with air where the pressure is 7.58 × 10⁵ Pa and the temperature is 25 °C. Assume that the average molar mass of air is 28.97 g/mol.

An ideal carbon monoxide (CO) gas has initially a temperature of 290 K and a pressure of 3 atm. The gas undergoes an expansion from 0.5 m³ to 1 m³. The CO final pressure is 4 atm. Carbon monoxide's molar mass is 28.0 g/mol. What is the mass (m) of CO?

A worker uses a hand air pump to inflate an air mattress. The worker notices that the volume of the mattress is 160 liters. The temperature of the air inside the mattress is 25°C, and the absolute pressure inside the mattress is 1.0272 atmospheres. What is the mass of air in the mattress?

The piston-cylinder assembly of an engine initially contains 0.5 mol of an ideal gas at a pressure of 1 atm. The gas inside the cylinder expands slowly and isothermally until the volume of the cylinder is doubled. What is the final pressure (p_f) inside the cylinder?

A balloon contains 7 × 10^-3 m³ of argon gas. The initial pressure and temperature are 1 atm and 290 K, respectively. The balloon is cooled until both its volume and pressure are halved. Find the final temperature (T_f).

An empty aerosol sprayer of volume 0.3 liters is filled with 0.3 g of carbon dioxide gas at a temperature of 25 °C. The molecular mass of carbon dioxide is 44 g/mol. Find the pressure (p) inside the sprayer. Express your answer i) in pascals and ii) in atmospheres.

A diver uses a scuba oxygen cylinder of 12 liters filled with 16 kg of diatomic oxygen gas (O₂). The molar mass of O₂ is 32.00 g/mol. How many moles of oxygen (n) are present in the cylinder?

Ethanol's data sheet shows that its critical density is 280 kg/m³. Ethanol density at room temperature is 789 kg/m³. Calculate the ratio (V_critical /V_{room temperature}) of the volume occupied by one mole of ethanol at the critical point to that at room temperature. Ethanol's molecular mass is 46 g/mol.

A cuboid-shaped container with a movable piston contains 0.0050 mol of gas at 1 atm pressure and 25°C. The piston is attached to a spring with a spring constant of 2000 N/m. If the piston is in equilibrium and the gas is heated to 75°C, how much is the spring compressed? Assume the cross-sectional area of the container is 8 cm².

A 10-cm-diameter cylinder with a movable piston contains helium gas at a temperature of 25°C and a pressure of 1 atm. The cylinder is oriented vertically, and the air above the piston is removed. The movable piston is made of aluminum and is 2 cm thick. When the gas temperature is increased to 100°C, the piston is found to have moved up to a position 15 cm above the bottom of the cylinder. What is the new pressure of the helium gas in the cylinder? Assume the density of aluminum is 2710 kg/m³ and the cross-sectional area of the cylinder is constant throughout.

A cube-shaped container with a side length of 8.0 cm with a movable piston contains carbon dioxide gas at a temperature of 27°C. The movable piston is made of steel and is 1.5 cm thick. At 27°C, the piston is found to have a position 5.0 cm above the bottom of the cube. The space above the piston is evacuated. What is the new equilibrium temperature of the gas when 3.5 J of heat energy has been transferred into the gas? Assume the density of steel is 8000 kg/m³.

A cylindrical container filled with chlorine gas is closed using a tight cork. The cork is used to reduce the volume of the gas to 3/5 of the original volume. Find the new mass density of the chlorine gas.

Dry ice sublimes to form a gas that occupies a volume V at 52°C. If the pressure remains constant and the temperature triples to 156°C, find the new volume. 

A sample of an ideal gas has a volume of V at 80°C. When the pressure of the gas remains fixed, find the volume when the Kelvin temperature is 1/5 of the initial Kelvin temperature. 

A sphere of diameter 220 mm holds 36 g of chlorine gas at room temperature (25°C). Calculate the number density of chlorine gas in the container. 

A spherical container of diameter 240 mm is filled with 34 g of nitrogen gas at 35°C. Determine the pressure of the nitrogen gas in the container as reported by a pressure gauge.

p₀, V₀, and T₀ are the starting state variables defining a certain sample of gas. The gas experiences isochoric cooling to a point p₁ = (2/5)p₀. Find V₁ (volume at p₁).

p₀, V₀, and T₀  describe the beginning state variables for a gas sample that experiences an isothermal expansion to a point  V₁ = 4V₀. Find T₁ (the temperature at V₁).

An ideal gas has a pressure p₀, volume V_0, and temperature T₀. The gas expands through an isothermal process to a point V₁ = 3V₀. Find P₁ (pressure at V₁).

A massless piston is used to cover a 320 mm diameter container. When a 25kg metal block is placed on the piston, the height of the gas in the container is 520 mm, and t = 285°C. The outer side of the piston is in contact with air at 1.0 atm pressure. If the gas is cooled to 5°C, find the new height of the gas. 

A 0.325 mol sample of argon is placed in an initially empty container with rigid non-elastic walls with a volume of 440 ml. The initial temperature of the gas is 25°C. The gas is heated using an oven to a final temperature of 450°C. Determine the pressure of the gas after heating.

A 125 mL drum holds 0.25 mol of helium gas at T = 15°C. The drum allows isochoric warming of the gas to T = 225°C. Use a pV graph with correct values on the axes to represent the process. 

0.25 mol of helium experiences the transformations shown in the figure below. Using the information given in the figure, find the temperature of the gas at point A.

An ideal gas of mass m follows the transformations shown in the figure below. Using the information given in the figure, find the pressure at point B.

n mole of an ideal gas in a piston–cylinder aperture experiences the transformations shown in the figure below. Calculate the volume at point C.

In a thermodynamic experiment, 5.0 g of nitrogen gas at 400 K is placed inside a piston cylinder of volume 2.0 × 10^-5 m³ under a pressure of 2.0 × 10⁵ Pa. First, the temperature of the gas is increased by 40% while its volume remains constant. Then the pressure is reduced by 30% at a constant temperature. Calculate the final volume of nitrogen gas.

Nitrogen storage tanks are cylindrical pressure vessels used to prevent product spoilage. A cylindrical pressure vessel containing 6.22 g of nitrogen gas is sealed with a lid to prevent leaks. The pressure inside the vessel is below 1.00 × 10⁵ Pa; thus, the lid is secured by the force exerted by the atmospheric pressure.  The vessel is 60cm long and has a 7.00 cm radius. If the force needed to pull the lid is equal to 400 N at a location where the atmospheric pressure is 1.10 × 10⁵ Pa, find the gas temperature.

A gas turbine engine has a cylindrical-shaped combustion chamber with a radius of 0.25 meters and a length of 1.0 meters. While operating, the air inside the chamber is at a temperature of 1800 degrees Celsius and a pressure of 1.6 MPa. Calculate the volume of air, at a temperature of 25 degrees Celsius and a pressure of 101.3 kPa, required to fill the combustion chamber.

An industrial machine, as illustrated below, uses compressed 0.23 moles of carbon dioxide gas to lift a square-shaped movable piston that weighs 38.0 kg. The side of the piston has a length of 15.0 centimeters. If the temperature of the carbon dioxide gas is 300 K, determine the value of L.

A circular piston could vertically move due to the expansion or compression of 0.15 moles of compressed nitrogen gas as shown below. What is the change in position, in meters, of the piston if the nitrogen temperature is increased from 700.0 °C to 800.0 °C?

A group of researchers prepared a spherically shaped inflatable weather balloon filled with helium gas at a temperature of 15.0°C and a pressure of .980 atm making a radius of 3.0 meters. As the balloon reaches the highest altitude, the temperature drops to -60.0°C, and the atmospheric pressure is at 0.010 atm. Determine the radius of the weather balloon at that altitude.

A gaseous substance at 650°C is treated through the processes represented graphically below. Identify the process.

An ideal gas originally at a temperature of 750°C follows the transformation below. Calculate the amount of the gas in moles.

A sample of solid iodine weighing 20 g is placed in a sealed container with a volume of 500 ml at 25°C. The container is then heated to 80°C, causing the solid iodine to sublime into a gas. What is the gas pressure inside the container at 80°C, assuming all of the solid iodine sublimates and the container remains sealed? Consider the molar mass of solid iodine is 254 g/mol.

A gas in a piston has an initial volume of 500 ml, a pressure of 1.5 atm, and a temperature of 25°C. The gas then undergoes isothermal compression, decreasing its volume to 250 mL. Next, the gas is cooled at a constant volume until the pressure is reduced to 1 atm. Draw this two-step process on a pressure-volume (pV) diagram.

A container is filled with 10 g of helium gas at a temperature of 25°C and a pressure of 2.5 atm. The volume of the gas doubles due to isobaric expansion. Determine the final volume of the gas.

Consider a container filled with 15 g of methane gas at a pressure of 2.2 atm and a temperature of 30°C. The gas undergoes an isobaric expansion, during which its volume increases by a factor of 4. Determine the final temperature of the gas in degrees Celsius.

Create a question depicting a practical scenario for the following equation: p₂ = 500 cm³/150 cm³× 1 × 3.5 atm

Create a problem statement for the situation represented by the equation:
(T₂+273) K = (300 kPa)/(700 kPa) × 1 × (350+273) K

In environmental science research, researchers use vacuum chambers to simulate the low-pressure environment of Earth's atmosphere. The gauge of a chamber reads 0.53 mmHg. Calculate the proportion of the standard atmospheric pressure that this pressure represents.

Construct a pressure vs volume diagram for the equation given below: V₂ = (550 + 273) K / (120+273) K × 1 × 350 cm³.

A buoy has a gauge pressure of 32 psi when the temperature is 25°C. The buoy has a volume of 10 liters and contains pure helium gas. Due to a puncture, the gauge pressure drops to 25 psi on a day when the temperature is 20°C. What mass of helium gas leaked out of the buoy assuming the volume remains constant?

A cylinder with a tightly held and frictionless piston of mass 1 kg is divided into two chambers by a fixed partition. The left chamber has a length of 20 cm and contains an ideal gas at a pressure of 1 atm and a temperature of 25°C. The right chamber contains a spring with a spring constant of 500 N/m, initially compressed by 5 cm. If the piston is moved 1.5 cm to the right, find the net force on the piston assuming the gas temperature remains constant.

A spherical container with a diameter of 50 cm is filled with helium gas at a temperature of 25°C and a pressure of 2 atm. Estimate the number of helium gas molecules in the container that have a speed between 500 m/s and 1000 m/s. Assume that the container is perfectly insulated and that the helium gas behaves as an ideal gas. Consider 2.2 % of helium gas molecules have a speed between 500 m/s and 1000 m/s.

Researchers in plasma physics use specialized plasma chambers operating at low pressures. The temperature and pressure inside a spherical chamber of radius 25 cm are 310 K and 0.095 mm Hg, respectively. Calculate the number of molecules inside the chamber.

Cows are part of the livestock industry, which contributes significantly to methane (CH4) emissions. According to studies, a cow produces 216 g of methane on average each day. Since methane gas is a major greenhouse gas, there is ongoing research to find ways to reduce the amount of methane gas produced by cows. Suppose that a system has been developed to store the emitted methane. What would be the length of a cylinder with a radius of 1.00 m needed to contain at the STP the monthly methane emissions of a cow? Assume each month has 30 days.

A teacher placed a cylinder with a length of 1.0 m and cross-sectional area A in contact with the surface of a large tank filled with oil of density 950 kg/m³. The upper part of the cylinder is sealed. She slowly moves the cylinder until its upper part is level with the oil's surface. The oil would start to fill the cylinder, compressing the air column inside. The oil inside the cylinder stops at a distance of h from its upper part, as shown below. Calculate h.

An open-bottomed container of height 2.5 m is submerged in the water of a lake to provide a temporary refuge for a scientist studying the lake's marine life. The container is initially filled with air at an initial temperature of 30 °C. The closed upper part reaches a depth of 30 m. What is the height the water in the container will reach once the air inside has reached a constant temperature of 20 °C?

A pressure cooker operates at a pressure of 2.0 atm and a temperature of 120°C.  A 1.0-liter bottle is used to collect the steam that is released from the safety valve. What is the amount of water, in liters, required to produce enough steam to fill the bottle? Assume that the steam inside the bottle is at 2.0 atm and 120 °C.

Before a race, a bike mechanic inflated a road bike tire to a pressure of 110 psi. The air temperature inside the tire was 15°C. At the end of the race, the air temperature inside the tire increased to 35°C. Calculate the tire's pressure at the end of the race.

A cylinder in an Otto cycle engine in a ship has a volume of 1828 liters. The cylinder sucks in the air from the atmosphere at sea level at a temperature of 15°C. Assuming the air contains 78% nitrogen, calculate the number of nitrogen molecules in the cylinder when the cylinder has sucked maximum capacity.

Charcoal fire can attain temperatures as high as 1200°C. The temperature of the gas escaping from the fire can be assumed to be at 1200°C, and a pressure of 1.013 × 10⁵ Pa. Calculate the approximate number density of gas particles in the gas escaping the fire.

A container designed to hold argon gas for welding applications has a base of 1.00 × 10  ^-2 m ². The container is fastened with a movable and impermeable piston, which is initially at a height of 0.50 m. The piston has a mass of 12 kg and is free to move up and down without any friction. The argon gas inside the container is maintained at a constant temperature of 25 °C using a thermostat. 10 kg of iron powder is carefully deposited onto the top of the piston, where it remains. Calculate the final height of the piston.

A group of students was conducting experiments on how far a green light can travel in the water before being completely scattered by water molecules. The students presumed that the mean free path of photon-water molecule collisions is equal to the distance that light can travel through water before it scatters completely. Using a very thin-walled transparent container and at 15.0°C, the distance the green light could travel through the water was 20.3 meters.  If the water is heated, increasing its temperature to 90.0°C while keeping its pressure constant, how far can green light travel before being completely scattered? Hint: The mean free path of the photons is inversely proportional to the product of the number density and the square of the radius of the water molecules. 

In the outer reaches of our solar system, a cloud of hydrogen gas is formed. The cloud has a density of 1 atom per 2.0 cm³ and a temperature of -271.0 degrees Celsius. What is the pressure inside this cloud? Assume hydrogen gas behaves like an ideal gas.

A beam of beta particles passes through a cylindrical chamber with a radius of 11.5 cm and a length of 80.0 cm filled with nitrogen gas. If the mean free path of the beta particles is measured to be 1.0 cm, what should be the pressure inside the chamber when the temperature is at -1.0 degrees Celsius? Let us assume the nitrogen atoms have a radius of 0.10 nm.

A 22.0 g sample of gaseous oxygen at 25.0℃ undergoes an increase in pressure from 1.00 atm to 16.0 atm at constant volume. Compute the final temperature.

Calculate the pressure in a cylinder with a volume of 70.0 L containing 210 kg of helium at 10.2°C.[The molar mass of helium is 4.0 g/mol.]

A tank full of helium used for filling balloons at a party store has a gauge pressure of 1500 psi or 10,342 kPa, and has a volume of 20 L or 0.020 m³ at temperature of 300 K. Imagine that the valve is opened accidently, and the helium started to leak at a rate of 3.0 L/min. Calculate how long the tank would last.

During an underwater expedition, an explorer inhales oxygen from her tank, filling her lungs to their maximum capacity of 5.5 L while she is located 10 m below the sea's surface. If she quickly ascends to the surface, what would be the volume of her lungs? Would this be a good idea to attempt? Hint: The density of seawater is 1025 kg/m³ and 1 atm = 101325 Pa.

A compressed air tank used in a paintball gun has an absolute pressure of 200 atmospheres when it is fully charged at a room temperature of 25°C. The tank has a capacity of about 12 liters. Assuming that the temperature remains constant during this process, what will be its new volume value when it expands to atmospheric pressure?

An industrial-grade compressor intakes approximately 7000.0 L of air at atmospheric pressure and compresses it into a storage vessel with an internal capacity of 70.0 L. This process occurs at an average temperature of 15.0°C. Calculate the number of moles of air contained in the storage vessel. [Hint: The gas constant R = 8.314 J/ mol·K and 1 atm = 101325 Pa.]

If you consider an average air molecule to have a diameter of around 0.29 nm, what percentage of the empty space do the air molecules occupy in your car? Assume that the average temperature on a typical day is 20° C and that the pressure in the car is equal to atmospheric pressure. [Hint: The Boltzmann constant is 1.38 × 10^-23 J/K and 1 atm = 101325 Pa.]

If on an extremely hot day, meteorologists recorded temperatures reaching upwards of 50°C, and that the relative humidity was recorded as being around 70 %, what would be the approximate value for the partial pressure of water vapor on this specific day, given that the saturated vapor pressure of water at the same temperature is 1.23 × 10⁴ N/m²?

In the study of gas molecules, a significant measure is the 'mean free path', representing the average distance a molecule travels between collisions. Using principles from thermodynamics, derive an expression for the mean free path in terms of temperature and pressure rather than molecular density (N/V). Subsequently, apply this expression to determine the mean free path for hydrogen molecules under conditions of 2.00 atm pressure and 500 K temperature. Consider the diameter of a hydrogen molecule 2.40 × 10^-10 m.

A weather balloon is filled with hydrogen (H₂) gas and can ascend to various altitudes. The hydrogen gas inside the balloon is maintained at a pressure consistently 1.1 times the ambient pressure at any altitude. What is the buoyant force on the balloon at any altitude 𝑦? Assume that the temperature of the hydrogen and air are the same.

A cubical room having side length 's' contains 'N' particles at temperature 'T' and under pressure 'P'. Find out how frequently these particles hit one face of the room. Also, express an approximation of this frequency using the mass 'm' of a particle. Hint: Consider the motion in one direction only, say along the x-axis.

In a parallel universe, the behavior of gases is different. All other laws are similar to that of ours except the pressure of an ideal gas is inversely proportional to the cube of its volume at constant temperature and mass. Find the ideal gas law in this universe.

In a hypothetical universe, the behavior of 'perfect' gases is somewhat different from ours. In this universe, if the pressure and mass are kept constant the volume is proportional to the absolute temperature raised to a power of 3/2. Given that, all other laws are similar to those of our universe, what would be the ideal gas law in this universe?

A theoretical physicist is considering a different universe in which the laws of physics are somewhat different than those of ours. In this universe, the ideal gas law is given by PV³ = n²RT^2/3. If at a temperature of 200 K and pressure of 1.5 atm, 1.0 mol of an ideal gas in this universe occupies a volume of 15 L, What would be the universal gas constant in this universe?

Using the van der Waals equation of state, find the formulas for the critical temperature and critical pressure. Hint: At the critical point, the pressure versus molar volume graph has an inflection point, which means the first and second derivatives with respect to molar volume are zero.

An ideal gas is maintained at a steady temperature. When it is stored in a container, demonstrate how the pressure is related to the Bulk modulus (B) of this perfect gas.

As the manager of a hot yoga studio with a volume of 10 m³ and a temperature of 40°C, you want to increase the relative humidity from 0% to 15% to enhance the experience for your yogis. Calculate how much water (in kg) needs to be evaporated to achieve this humidity level.

Consider a free diver holding their breath while diving to a depth of 5.0 m below the surface of the sea. what is the approximate fractional change in the free diver's lung volume when they reach a depth of 5.0 m? Assume that, in equilibrium, the air pressure in the diver's lungs matches that of the surrounding water pressure. Use: sea water density 𝜌=1025 kg/m³ and atmospheric pressure 𝑃₀=1.013×10⁵ Pa.

A diatomic gas undergoes a thermodynamic cycle in a heat engine, as illustrated in the 𝑃𝑉 diagram below. Using the ideal gas law, determine the number of moles that are present in this gas.

A diatomic gas undergoes a thermodynamic cycle in a heat engine, as illustrated in the 𝑃𝑉 diagram below. Using the ideal gas law, determine the temperature at point $iii$.

At STP, 2.00 moles of helium gas undergoes isothermal expansion so that the volume at point B is 3.2 times the volume at point A. The pressure drops to point C as heat is extracted at a constant volume. The gas goes back to its original state (A) through adiabatic compression. What are the pressures at points B and C? Assume helium gas behaves like an ideal gas.

At STP, 2.00 moles of helium gas undergoes isothermal expansion so that the volume at point B is 2.2 times the volume at point A. The pressure drops to point C as heat is extracted at a constant volume. The gas goes back to its original state (A) through adiabatic compression. What is the temperature at point C? Assume helium gas behaves like an ideal gas.