The Ideal Gas Law: Density: Videos & Practice Problems

Density, a fundamental concept in chemistry, measures the mass of a substance per unit volume. For gases, which are less dense than solids and liquids, density is typically expressed in grams per liter. The ideal gas law, represented by the equation:

$PV=nRT$

can be adapted to calculate the density (D) of a gas at specific pressure (P) and temperature (T) conditions. By manipulating the ideal gas law, we derive the formula for density as:

$D=\frac{P\mathrm{MM}}{RT}$

where MM is the molar mass, R is the ideal gas constant, and T is the temperature in Kelvin. This relationship is crucial for understanding the behavior of gases under varying conditions and can be remembered by the mnemonic "Dreams push me over rough times," symbolizing the equation components. This formula is essential for predicting how gases will respond to changes in pressure and temperature, which is critical for applications in various scientific and industrial processes.

The Ideal Gas Law: Density

Density is a fundamental concept in science that quantifies how much mass is contained in a given volume. The formula for density can be expressed as:

\( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \)

In this equation, mass is measured in grams (g), and volume is typically measured in liters (L) when dealing with gases. This distinction is important because gases have significantly lower densities compared to solids and liquids. Therefore, the appropriate unit for gas density is grams per liter (g/L), rather than grams per milliliter (g/mL) or grams per cubic centimeter (g/cm³).

To summarize, when calculating the density of a gas, ensure that you use the mass in grams and the volume in liters, reflecting the unique properties of gases in comparison to other states of matter.

Gas Density Example

To determine the mass of an unknown gas sample with a density of 1.70 grams per liter and a volume of 120 milliliters, we start by converting the volume from milliliters to liters. Since 1 milliliter is equal to \(10^{-3}\) liters, we can convert 120 milliliters as follows:

\[120 \, \text{ml} = 120 \times 10^{-3} \, \text{liters} = 0.120 \, \text{liters}\]

Next, we apply the density formula, which relates mass, volume, and density:

\[\text{Density} = \frac{\text{Mass}}{\text{Volume}}\]

Rearranging this formula to solve for mass gives us:

\[\text{Mass} = \text{Density} \times \text{Volume}\]

Substituting the known values into the equation:

\[\text{Mass} = 1.70 \, \text{grams/liter} \times 0.120 \, \text{liters} = 0.204 \, \text{grams}\]

When considering significant figures, the density (1.70 grams per liter) has three significant figures, while the volume (0.120 liters) has four. Therefore, the final answer should be reported with three significant figures, resulting in a mass of:

\[\text{Mass} = 0.204 \, \text{grams}\]

This calculation demonstrates how to effectively use density to find mass, ensuring proper unit conversions and adherence to significant figure rules.

Density from Ideal Gas Law

The ideal gas law is a fundamental equation in chemistry that relates the pressure, volume, temperature, and number of moles of a gas. A useful derived version of this law allows us to calculate the density of a gas under specific conditions of pressure and temperature. The formula for density can be expressed as:

\( d = \frac{P \cdot M}{R \cdot T} \)

In this equation, \( d \) represents density, \( P \) is pressure, \( M \) is the molar mass of the gas, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin. To help remember the order of these variables, you can use the mnemonic "dreams push me over rough times," where each word corresponds to a component of the formula: "dreams" for density, "push" for pressure, "me" for molar mass, "over" for the division, and "rough times" for the product of \( R \) and \( T \).

This mnemonic serves as a handy tool for recalling how to apply the ideal gas law in the context of density calculations. Understanding this relationship is crucial for solving various problems in gas behavior and thermodynamics. Once you are comfortable with this formula, you can proceed to practice its application in different scenarios to reinforce your learning.

Density Determination from Ideal Gas Law

The derivation of a new version of the ideal gas law begins with the fundamental concept of density, defined as the ratio of mass to volume, expressed mathematically as:

Density (d) = &frac{m}{v}

In the context of the ideal gas law, we incorporate molar mass, which can be represented by the equation:

Molar Mass (M) = &frac{mRT}{PV}

Here, R is the ideal gas constant, T is the temperature in Kelvin, P is the pressure, and V is the volume. By substituting the density formula into the ideal gas law, we can express molar mass in terms of density:

M = &frac{dRT}{P}

To isolate density, we can manipulate the equation using algebra. By multiplying both sides by pressure, we have:

M * P = d * R * T

Next, we isolate density by dividing both sides by RT:

d = &frac{MP}{RT}

This final equation shows how density can be expressed in terms of molar mass, pressure, and temperature. Understanding this derivation is crucial for applying the ideal gas law in various scenarios, particularly in calculating the density of gases under different conditions.

Molecular Formula Example

To determine the molecular formula of a gaseous compound composed of nitrogen and hydrogen, we can utilize the provided density, pressure, and temperature values. The density of the gas is 0.977 grams per liter, the pressure is 0.69474 atmospheres, and the temperature is 373.15 Kelvin. These parameters allow us to calculate the molar mass of the gas using the ideal gas law in its density form.

The formula for molar mass (\(M\)) derived from the ideal gas law is:

\(M = \frac{dRT}{P}\)

Where:

• d = density of the gas (0.977 g/L)
• R = ideal gas constant (0.0821 L·atm/(K·mol))
• T = temperature in Kelvin (373.15 K)
• P = pressure in atmospheres (0.69474 atm)

By substituting the known values into the equation, we can isolate the molar mass:

\(M = \frac{(0.977 \, \text{g/L}) \times (0.0821 \, \text{L·atm/(K·mol)}) \times (373.15 \, \text{K})}{0.69474 \, \text{atm}}\)

Calculating this gives:

\(M \approx 43.06 \, \text{g/mol}\)

With the molar mass determined, the next step is to compare this value to the molar masses of potential nitrogen and hydrogen compounds. The goal is to find a compound whose molar mass is closest to 43.06 g/mol. Upon comparison, the compound that matches this molar mass is option C, which has a molar mass of approximately 43.038 g/mol.

In summary, by using the density, pressure, and temperature of the gas, we can effectively apply the ideal gas law to find the molar mass, which then allows us to identify the molecular formula of the compound.

Consider two containers of gases at the same temperature. One has helium at a pressure of 1.00 atm. The other contains carbon dioxide with the same density as the helium gas. What is the pressure of the carbon dioxide gas sample?

• A. 0.063 atm
• B. 0.091 atm
• C. 0.133 atm
• D. 1.51 atm
• E. 2.71 atm

Answer: (B)

Determine the molecular formula of a gaseous compound that is 49.48% carbon, 5.19% hydrogen, 28.85% nitrogen, and 16.48% oxygen. At 27°C, the density of the gas is 1.5535 g/L and it exerts a pressure of 0.0985 atm.

• A. C₂₀H₄₄N₈O₄
• B. C₁₆H₂₀N₆O₂
• C. C₁₄H₂₂N₂O₂
• D. C₁₆H₂₀N₈O₄

Answer: (D)

Here’s what students ask on this topic:

How can density be found using the ideal gas law?

To find the density of an ideal gas using the Ideal Gas Law, you need to start with the law's formula, which is:

$PV=nRT$

Here, P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin.

Density (ρ) is defined as mass (m) per unit volume (V), so:

$\rho =\frac{m}{V}$

To find the mass of the gas, you can use the molar mass (M) of the gas and the number of moles (n):

$m=nM$

Substitute the expression for m into the density equation to get:

$\rho =\frac{nM}{V}$

Now, rearrange the Ideal Gas Law to solve for n/V:

$\frac{n}{V}=\frac{P}{RT}$

Substitute n/V in the density equation with P/RT to get the final expression for density:

$\rho =\frac{MP}{RT}$

So, the density of an ideal gas can be found using the equation:

$\rho =\frac{MP}{RT}$

where M is the molar mass of the gas, P is the pressure, R is the ideal gas constant, and T is the temperature in Kelvin.

How can density be found using the ideal gas law?

Density can be found using the ideal gas law, which is expressed as:

$PV=nRT$

Here, P stands for pressure, V is volume, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature in Kelvin.

To find the density (ρ), you need to rearrange the ideal gas law to solve for mass (m) over volume (V), since density is mass per unit volume (ρ = m/V). You can replace n (the number of moles) with m/M (mass over molar mass) in the ideal gas law equation:

$PV=\frac{m}{M}RT$

Now, solve for m/V, which is the density (ρ):

$\rho =\frac{PM}{RT}$

This equation shows that the density of an ideal gas is directly proportional to its pressure (P) and molar mass (M), and inversely proportional to the temperature (T) and the ideal gas constant (R). By knowing the pressure, temperature, and molar mass of the gas, you can calculate its density.