The Ideal Gas Law: Molar Mass: Videos & Practice Problems

Understanding the molar mass of a gas is crucial in chemistry, and the ideal gas law provides a method to calculate it. Molar mass (M) is the mass of a substance (in grams) divided by the amount of substance (in moles). The ideal gas law can be rearranged to find the molar mass of a gas using the formula:

$M=\frac{mRT}{PV}$

where m is the mass of the gas in grams, R is the ideal gas constant, T is the temperature in Kelvin, P is the pressure, and V is the volume. This formula allows for the direct calculation of molar mass from the properties of a gas, streamlining the process and eliminating the need for step-by-step calculations. It's a powerful tool for chemists to determine the molar mass from experimental gas data, providing insight into the molecular composition and behavior of gases.

The Ideal Gas Law: Molar Mass

Understanding molar mass is essential in the study of gases, particularly when applying the ideal gas law. Molar mass is defined as the mass of a substance divided by the amount of that substance, typically expressed in grams per mole (g/mol). In this context, we denote molar mass with the symbol M, the mass of the gas in grams as m, and the amount of gas in moles as n.

The relationship can be summarized by the equation:

M = \frac{m}{n}

Where M is the molar mass, m is the mass of the gas, and n is the number of moles. This equation allows us to calculate the molar mass of a gas when we know its mass and the number of moles present.

By integrating this concept with the ideal gas law, which is expressed as:

PV = nRT

where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature in Kelvin, we can derive the molar mass of a gas under specific conditions. This application of the ideal gas law provides a practical method for determining the molar mass, enhancing our understanding of gas behavior in various scenarios.

Molar Mass 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. It can be expressed as:

PV = nRT

Where:

• P = pressure of the gas
• V = volume of the gas
• n = number of moles of the gas
• R = ideal gas constant
• T = temperature in Kelvin

In addition to finding the number of moles, the ideal gas law can be adapted to calculate the molar mass of a gas. The molar mass (M) can be derived from the ideal gas law by rearranging the equation. The formula for molar mass can be expressed as:

M = \frac{mRT}{PV}

In this equation:

• M = molar mass of the gas
• m = mass of the gas sample
• R = ideal gas constant
• T = temperature in Kelvin
• P = pressure of the gas
• V = volume of the gas

A mnemonic to remember this formula is: "Molar mass really tests our valuable patients," where each word corresponds to the components of the equation. This approach allows for a straightforward calculation of molar mass without needing to break down the process into smaller steps.

Understanding this adaptation of the ideal gas law is essential for solving problems related to gas behavior and properties in various scientific contexts.

Ideal Gas Law Derivation

The ideal gas law is a fundamental equation in chemistry that relates the pressure (P), volume (V), number of moles (n), ideal gas constant (R), and temperature (T) of a gas. The standard form of the ideal gas law is expressed as:

PV = nRT

To derive a new version of the ideal gas law that incorporates molar mass, we start with the definition of molar mass (M), which is the mass (m) of a substance divided by the number of moles (n):

M = \frac{m}{n}

By rearranging this formula to isolate the number of moles, we multiply both sides by n:

m = nM

Next, we can express the number of moles as:

n = \frac{m}{M}

Substituting this expression for n back into the ideal gas law gives us:

PV = \left(\frac{m}{M}\right)RT

To isolate molar mass (M), we first multiply both sides by M:

PV \cdot M = mRT

Then, we divide both sides by PV to solve for molar mass:

M = \frac{mRT}{PV}

This derived equation shows how molar mass can be calculated using the mass of the gas, the ideal gas constant, the temperature, and the pressure and volume of the gas. Remembering this relationship can be helpful in solving problems related to gases and their properties, reinforcing the connection between molar mass and the ideal gas law.

Gas Identity Determination using Molar Mass Example

In this example, we are tasked with identifying an unknown gas based on its mass, volume, pressure, and temperature. The gas has a mass of 0.1727 grams and is contained in a 125 milliliter flask at a pressure of 0.833 atmospheres and a temperature of 20 degrees Celsius. To find the identity of the gas, we will calculate its molar mass using the ideal gas law.

The formula for molar mass derived from the ideal gas law is given by:

M = \frac{RT}{PV}

Where:

• M is the molar mass in grams per mole (g/mol)
• R is the ideal gas constant, 0.08206 L·atm/(mol·K)
• T is the temperature in Kelvin (K)
• P is the pressure in atmospheres (atm)
• V is the volume in liters (L)

First, we need to convert the temperature from Celsius to Kelvin:

T = 20 + 273.15 = 293.15 \, K

Next, we convert the volume from milliliters to liters:

V = 125 \, mL = 0.125 \, L

Now we can substitute the known values into the molar mass equation:

M = \frac{(0.08206 \, \text{L·atm/(mol·K)}) \times (293.15 \, K)}{(0.833 \, \text{atm}) \times (0.125 \, L)}

Calculating this gives us:

M \approx 39.90 \, g/mol

To identify the gas, we compare the calculated molar mass with the known molar masses of various gases:

• N₂ (Nitrogen) = 28.02 g/mol
• Ar (Argon) = 39.95 g/mol
• O₂ (Oxygen) = 32 g/mol
• Ne (Neon) = 20 g/mol
• CH₄ (Methane) = 16 g/mol

The closest match to our calculated molar mass of approximately 39.90 g/mol is Argon (Ar), which has a molar mass of about 39.95 g/mol. Thus, the identity of the unknown gas is Argon.

To identify a homonuclear diatomic gas, a chemist weighted an evacuated flask with a volume of 3.9 L then filled it with the gas at a pressure of 2.00 atm and 29.0 ºC. The chemist then re-weighted the flask and recorded the difference in mass as 8.81 g. Identify the gas.

• A. H₂
• B. N₂
• C. Cl₂
• D. F₂
• E. O₂

Answer: (B)

What is the molecular formula of a compound that contains 39.0% carbon, 16.0% hydrogen, and 45.0% nitrogen, if 0.1576 g of the compound occupies 125 mL with a pressure of 0.9820 atm at 295.15 K?

• A. CH₃N
• B. CH₅N
• C. C₂H₃N₂
• D. C₂H₅N₂

Answer: (B)

Here’s what students ask on this topic:

How can molar mass be found using the ideal gas law?

To find the molar mass using the Ideal Gas Law, you first need to know the other variables in the equation PV=nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature.

Assuming you have the values for pressure, volume, and temperature, and you know the amount of gas in grams (m), you can rearrange the Ideal Gas Law to solve for n (moles), which is m (mass of the gas in grams) divided by M (molar mass of the gas in grams per mole):

n=PVRT

Now, to find the molar mass (M), rearrange the equation to solve for M:

M=mRTPV

Here's a step-by-step guide:

1. Measure the mass of the gas in grams (m).
2. Measure the volume (V) of the gas.
3. Measure the temperature (T) in Kelvin.
4. Measure the pressure (P) of the gas.
5. Use the ideal gas constant (R) in the appropriate units to match your pressure and volume units.
6. Plug these values into the rearranged Ideal Gas Law equation to solve for the molar mass (M).

Remember to keep your units consistent, especially for temperature (which should be in Kelvin).

How can I find molar mass using the ideal gas law?

To find the molar mass using the ideal gas law, you'll need to rearrange the ideal gas law formula to solve for molar mass. The ideal gas law is:

P V = n R T

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

First, you need to isolate n (the number of moles) by rearranging the equation:

n = P V R T

Then, remember that molar mass (M) is the mass (m) of the gas divided by the number of moles (n):

M = m n

Combine these two relationships by substituting n from the ideal gas law:

M = m P V R T

Now you have a formula for molar mass that incorporates the ideal gas law:

M = m R T P V

To find the molar mass, you'll need to know the mass of the gas sample (m), the pressure (P), the volume (V), and the temperature (T). Insert these values into the formula, and use the appropriate value for R (8.314 J/(mol·K) for SI units), and you can calculate the molar mass of the gas.