Heat Equations for Special Processes & Molar Specific Heats: Videos & Practice Problems

A tube is filled with 0.150 moles of an ideal monoatomic gas with a pressure (initial) of 1.01 × 10⁵ Pa. The initial volume of the gas is 0.00420 m³. The tube expands to a final volume that is three times the original volume. If the process is adiabatic, determine the final temperature (in K) and pressure.   

A balloon of volume 0.00320 m³ contains an ideal gas at a pressure of 1.20 × 10⁵ Pa. The number of moles in the ballon is 0.120. The balloon expands to 1.8 times the original volume. Calculate the final temperature (in K) and pressure assuming an isobaric process.

A container with a movable piston has 0.220 moles of helium.  The initial pressure of helium is 1.01 × 10⁵ Pa. The volume of the gas at this pressure is 0.00575 m³. The gas undergoes a process where the final volume is 1.5 times the initial volume. Determine the final temperature (in K) and pressure if the process is isothermal.

A  hot air balloon contains air at a pressure equal to atmospheric pressure. A heater is used to raise the temperature of the air in the balloon to a temperature of 95.0°C at ground level (P = 1 atm). The balloon is allowed to rise in the atmosphere. The process taking place in the hot air as the balloon rises in the atmosphere can be considered adiabatic. Determine the temperature of the air in the balloon when the surrounding pressure is 0.75 atm. Treat air like an ideal gas with γ = 1.40. Ignore the effects of the balloon on the enclosed air. 

A 0.120 m³ container is filled with an ideal diatomic gas at a pressure of 1.80 × 10⁵ Pa. The gas is compressed adiabatically to a final volume of 0.0600 m³. Determine the ratio final to the initial temperature, (T_f/T₀). Does the process heat or cool the gas?

Compressed air in a container fitted with a piston is used as a shock absorber. In one instant, the air is compressed to 75% of its initial volume as it absorbs the shock. The shock absorber initially contained inert N₂ gas at a pressure of 4.50 atm and 25.0°C. The container is a cylinder of diameter 0.180 m and height 0.420 m. Determine the temperature of N₂ at maximum compression if the process is adiabatic and N₂ is an ideal gas. 

Piston air compressors are devices used to compress air to high pressures for storage in tanks. The device takes in air at atmospheric pressure (1 atm) and 18.0 °C. The compression is very fast such that it is assumed adiabatic. If the final volume is 0.120 of the initial volume, and air behaves like an ideal gas with γ = 1.40, determine the temperature and pressure at this instant of the compression. 

A sample of helium has an initial volume of 0.140 m³ and initial pressure of 2.45 × 10⁵ Pa. It is compressed adiabatically to a final volume of 0.0700 m³. Calculate the final pressure. Treat helium like an ideal gas, helium is monoatomic. 

A container fitted with a piston contains 6.50 moles of an ideal monoatomic gas. The original pressure and volume are 3.20 × 10⁴ Pa and 0.850 m³ respectively. The gas is allowed to expand adiabatically, doing 1640 J of work. Calculate the final pressure following the expansion.

0.840 mol of an ideal gas with an initial pressure of 1.60 atm is taken through an isothermal process at 82.0°C. The volume is lowered to 40.0% of the original volume. Does heat flow between the gas and the surroundings occur? If yes, how much and in which direction (use positive sign if into the gas; negative sign if out of the gas)?

The volume of an ideal gas in a metallic container is reduced at a uniform temperature. In the process, the gas loses 560 J of heat to keep its temperature uniform. Determine the work done by the gas in the process.

Consider a cylinder that holds 0.50 moles of helium gas at a pressure of 3.0 atm. Calculate the rise in temperature if 75 J of heat energy is absorbed by the gas during the isochoric process.

Consider a system containing a helium gas that undergoes the process A→B→C as described in the figure below. Determine the heat needed or removed for i) process A→B and ii) process B→C.

2.0 g of nitrogen gas at a pressure of 5.0 atm is filled in a container. Determine the quantity of heat that should be supplied to raise the temperature of the gas by 120°C keeping the pressure constant.

Consider a sealed cylindrical container with a movable piston that contains 0.500 moles of nitrogen gas at a pressure of 2.00 atm and a temperature of 25.0°C. The container has an initial length of 10.0 cm and the piston is fixed in place. If 1500 J of heat energy is added to the system, what is the resulting final length of the container at constant pressure?

In a physics demonstration, an instructor assembled an apparatus with two vertically aligned thermally insulated compartments separated by a thin barrier. The upper compartment contains 0.050 mol of xenon gas at an initial temperature of 500 K, and the lower compartment holds 0.025 mol of xenon gas at an initial temperature of 250 K. The lower compartment is ingeniously connected to a vertical barrel fitted with a 3.0 kg piston of radius 6.0 cm and the air pressure above the barrel is maintained at 1.5 atm. The piston is designed to move seamlessly up/down the barrel without friction. The dimensions of the barrel and the volumes of the compartments remain unspecified. Determine the amount of heat transferred between the upper and lower compartments during the instructor's demonstration. 

Consider a sample of helium gas with a mass of 7.0 g, an initial pressure of 2.0 atm and an initial temperature of 25°C. Helium is compressed at constant pressure until its volume is reduced to one-third of its initial value. Determine the final temperature of the gas.

Consider a sealed container with a movable piston that contains 0.10 moles of carbon dioxide gas at an initial pressure of 4.0 atm and an initial temperature of 300 K. By maintaining a constant pressure, carbon dioxide is expanded until its volume doubles. How much energy must be added to the system to cause this expansion?

A gas is contained in a piston and cylinder assembly at an initial pressure of 2.0 atm and a volume of 500 mL. The gas undergoes an isochoric cooling until its temperature is reduced to half of its initial value. Determine the final pressure of the gas.

How much heat energy is transferred from a 4.00 g of oxygen gas that initially has a volume of 2.50 liters and a pressure of 4.0 atm, and is then compressed at constant pressure until the volume is reduced by half, and finally undergoes an isochoric process until the gas reaches its original temperature of 25.0°C?  

Two polyatomic gases are mixed together. The composition is as follows: gas 1 has n₁ moles and degrees of freedom, f = 7, while gas 2 has n₂ moles and degrees of freedom, f = 9. Find an expression that can be used to calculate the specific heat at constant volume for 1 mole of gas.

0.042 moles of carbon dioxide gas undergo a thermodynamic process described by a pV-diagram below. Calculate the amount of heat released by the gas.

A gas follows a process where non-constant pressure follows the relation p = aV², where a is a proportionality constant. Calculate the work done in moving the gas from V₀ to 3V₀.

At an initial temperature of T₁ and volume V₁, a balloon contains n moles of a gas. The balloon is gradually inflated isothermally until its volume has tripled. Express the final temperature T₂ in terms of n, T₁, and V₁.

An experiment is conducted by a group of engineers to investigate the behavior of an ideal gas. They begin with n moles of the gas in a sealed container, at a temperature of T₁, and a volume of V₁. They expand the container isothermally, keeping the temperature constant until the volume tripled. Calculate the work done on the gas during this process in terms of n, T₁, and V₁.

An ideal gas is used as fuel in a high-tech engine. The engine starts with m moles of the gas at an initial temperature of T_i and a volume of V_i. During the engine's operation, the gas undergoes an isothermal expansion process, maintaining a constant temperature while the volume triples. Determine the amount of heat energy transferred to the gas during this process in terms of m, T_i, and V_i.

The pressure of a gas sample follows the relation p = aV^0.5, where a is a constant. If 0.120 moles of the gas at a starting temperature of 20°C  expands from 0.004 m³ to 0.009 m³ following the relation, calculate the work done by the gas.

An ideal gas is held within an inflexible chamber while it cools down and slowly loses about 500 kJ of heat energy.

(a) How much work has been performed throughout this procedure?

(b) What is the alteration in internal energy for this gas throughout this procedure?

A football stadium with a capacity of 24,000 m³ is packed with 1800 spectators for a game. Assuming there is no ventilation and the initial temperature is 293K, calculate the temperature rise of the air after 3.0 hours, given that each person generates heat at a rate of 80 W.

Calculate (i) the change in internal energy (ΔU), (ii) the work done by the gas (W), and (iii) the heat added to the gas (Q) if at a constant pressure of 1.00 atm, a 2.50-mole sample of O₂ gas undergoes heating from an initial temperature of 10°C to 200°C. Consider molar specific heat capacity at constant volume for O₂ gas, C_v = 29.1 J/mol·K.

Consider a cylinder with a movable piston containing n moles of an ideal gas. The gas is initially at a temperature T₁ and undergoes an adiabatic compression by the piston, resulting in a final temperature T₂. Determine the work done on the gas (W) during this adiabatic compression.

Consider a gas cylinder that is filled with nitrogen gas at a temperature of 20°C under normal atmospheric conditions (pressure roughly equal to 1.0 atm). The cylinder has dimensions as follows: height equals 40 cm and diameter equals 6.0 cm. If you were to block off its valve and then quickly press down on it until you have reduced its internal volume to one-third of its initial volume, what would be its final temperature? Assume the process is adiabatic.

In an Otto cycle engine, during compression stroke, air is taken into the cylinder at an initial temperature of 25°C (room temperature) and volume V_i. It's then adiabatically compressed till it reaches a peak temperature of about 700°C with the final volume V_f. The specific heat ratio γ for this process is 7/5. Determine what would be the value of V_i / V_f.

An ideal gas is allowed to adiabatically expand slowly in a piston. Its initial pressure and volume are P_i and V_i, and its final pressure and volume are P_f and V_f respectively. Given that the infinitesimal work done by the gas is dW = P.dV and for the quasistatic adiabatic process, PV^γ = constant, evaluate the expression for the total work done.

A certain amount of ideal gas undergoes a two-step transformation. Firstly, the heat flows into it while keeping its volume constant, causing an increase in pressure from 0.6 atm to 0.9 atm. Then, the system contracts, maintaining this newfound pressure, decreasing its volume from 0.9 L to 0.8 L. This causes the temperature to reach its initial value before the start of the transformations. Calculate (a) the total amount of work performed by this system, (b) the change in internal energy during these transformations, and (c) the overall heat flow.

For many substances at very low temperatures, Debye’s law states that their molar specific heat varies as per this relation: C = k(T³/T_D³). Given that for quartz crystal T_D = 945 K and k = 2205 J/mol·K, calculate how much heat is needed to raise the temperature of 1.5 mol of quartz crystal from initial temperature T_i = 20 K up to final temperature T_f = 40 K.

Imagine that you have a container with an ideal monoatomic helium gas of quantity equal to four moles at standard temperature pressure (STP). If this helium gets compressed slowly in an isothermal manner until it reaches one-fifth of its original volume then allowed to expand rapidly in an adiabatic manner back into its original size. Determine the highest & lowest temperatures & pressures attained by this helium and show on a PV diagram where these values occur. Use universal gas constant R = 0.0821 L.atm/K.mol.

In a laboratory, a scientist conducts a thermal equilibrium experiment. The scientist has a double-walled container with a thin metallic partition. On one side of the partition, there are 45 grams of liquid water (H₂O) at a temperature of 25°C. On the other side, there is a 5000 cm³ compartment filled with 0.50 mol of a monatomic gas initially at 12atm pressure. The partition allows for effective thermal contact between the liquid and the gas, while the double-walled container ensures proper insulation from the external environment. After a substantial amount of time has passed, what will be the new gas pressure? Assume that the container itself has negligible mass and does not influence the final result.

Two compressed gas cylinders, one containing neon and the other containing xenon, are brought into contact with each other. The neon cylinder has a volume of 12 L and is at a temperature of 300 K and a pressure of 2.5 atm. The xenon cylinder has a volume of 7.5 L and is at a temperature of 400 K and a pressure of 2.0 atm. Assuming the two cylinders are thermally insulated from the surrounding environment, calculate the amount of energy transferred between the two cylinders and the direction of the energy flow.

Real gases can approach the behavior of ideal gases under certain conditions. Helium is thought to be the closest to an ideal gas with γ = 1.660.  Calculate the molar heat capacities i) C_V and ii) C_p of helium.

A heat engine contains two moles of an ideal gas confined in a cylinder with a moveable piston. Fuel combustion provides 873 J of heat that increases the gas temperature from 45 °C to 60 °C under constant pressure. The heated gas expands and does 249 J of work on the piston. Calculate the ratio of heat capacities (γ) for the gas.

A student heats a vessel enclosing argon gas. During this process, argon absorbs an amount Q₀ of heat energy and expands at constant pressure. Find the percentage of heat energy involved in the expansion work of argon.

Equal samples (0.25 mol) of an ideal diatomic gas are treated through the processes represented below. Determine the amount of heat required in each process.

Interpret the information contained in the equation below that was used to solve a certain problem:

50 J = −(0.0142 mol)(8.31 J/mol K)(T)Ln(1/5).

Turbo boost allows heat engines to take in air at pressures above atmospheric pressure. A 0.300 L cylinder takes in air at 1.60 atm and 25 °C and 650 J of work is done compressing the air adiabatically. Find the final temperature and volume of the air in the cylinder.

At an initial pressure of 4.0 atm and a temperature of 200°C, 0.15 mol of N₂ gas is present in a closed vessel. Following an adiabatic expansion process, the volume of the gas triples. Determine the final pressure of the nitrogen gas within the vessel.

An adventurer brings an insulated, airtight cylinder containing 0.15 mol of argon gas (Ar) to assist in inflating a portable shelter. The initial temperature and pressure of the gas are 120°C and 2.5 atm, respectively. As the climber inflates the shelter, the gas expands adiabatically until the pressure drops by forty percent. Determine the argon gas's final temperature.

On a high-altitude weather balloon, a strong, insulated container holds a sample of gas. "As the balloon descends to a lower altitude, the external pressure changes, causing the volume of the gas to decrease to one-third of its initial value, and the pressure to increase by a factor of 3. In this case, determine the specific heat ratio of the gas in the container.

2.5 g of H₂ gas at room temperature and pressure undergoes adiabatic compression to 15 atm. Express the final volume in terms of the initial volume, V_f = _?_ V_i.

The volume of a sample of helium changes adiabatically from V₀ to 1.5V₀, where V₀ is the initial volume. Determine if the thermal energy of the helium sample increases/decreases and by what factor. If there is no change, state why. 

Neon gas expands adiabatically from V₀ to 3V₀, where V₀ is the initial volume. Find the factor by which C_v (molar specific heat at constant volume) is increased/decreased. If not changed, state why.

You are provided with two gas samples; a polyatomic (Cp/Cv = 4/3) and a monoatomic gas (Cp/Cv = 5/3). You are asked to mix the two gases to yield a gas with Cp/Cv = 7/5. Calculate the percentage of polyatomic molecules in the mixture.

If 0.0220 moles of argon are heated under conditions of constant volume and constant pressure from 15.0°C to 82.0°C, determine the change in internal energy during each process. Compare and explain the results. Hint: C_v = 12.47 J/mol•K, γ = 1.67.

If 0.0130 moles of neon are heated under conditions of constant volume and constant pressure from 17.0C to 80.0°C, the heat required is different. i) Why are the two values different? ii) Which process requires more heat? ii) How does this process use the extra heat?

A metallic container with a movable piston is filled with 0.0250 moles of argon at 20.0°C. If the temperature is increased to 70.0°C at constant pressure, how much heat is used? Sketch a PV diagram for the process. Hint: Cv = 12.47 J/mol•K, γ = 1.67.

A metallic container with a movable piston is filled with 0.0250 moles of argon at 18.0°C. If the temperature is increased to 78.0°C at constant volume, how much heat is used? Sketch a PV diagram for the process. Hint: C_v = 12.47 J/mol•K, γ = 1.67.

Determine the final pressure and temperature of a 2.00-mol sample of an ideal diatomic gas that initially starts at 2.00 atm and 25°C, after it undergoes an adiabatic expansion to twice its initial volume. Assume no molecular vibration.