The density of a gas can be calculated from its partial pressure using a modified version of the ideal gas law. The core equation is d = MP/RT, where:
- d is the density of the gas (g/L)
- M is the molar mass of the gas (g/mol)
- P is the partial pressure of the gas (atm)
- R is the ideal gas constant (0.0821 L·atm/mol·K)
- T is the temperature (K)
Explanation of the Equation
The equation d = MP/RT is derived from the ideal gas law, PV = nRT, where:
- P is pressure
- V is volume
- n is number of moles
- R is the ideal gas constant
- T is temperature
By rearranging the ideal gas law and using the relationship between moles, mass and molar mass (n=m/M), the density equation is achieved:
- Start with the ideal gas law equation: PV = nRT
- Rearrange to solve for moles per unit volume (n/V): n/V = P/RT
- Recall that n (moles) = m (mass) / M (molar mass). Substitue this into the equation: (m/M)/V = P/RT
- Rearrange to isolate m/V. m/V = MP/RT
- Realize that m/V is equivalent to density (d), so d = MP/RT
Using Partial Pressure
When dealing with a mixture of gases, you use the partial pressure of the specific gas you're interested in when calculating its individual density. The partial pressure of a gas is the pressure that gas would exert if it occupied the same volume alone.
Step-by-Step Calculation
- Identify the Gas: Determine the gas for which you are calculating the density, and obtain its molar mass (M).
- Obtain the Partial Pressure (P): Get the partial pressure of the gas in atmospheres (atm). If given in other units, convert to atm using appropriate conversion factors.
- Determine the Temperature (T): Make sure the temperature is in Kelvin (K). If given in Celsius or Fahrenheit, convert to Kelvin.
- Kelvin = Celsius + 273.15
- Use the Ideal Gas Constant (R): Utilize R = 0.0821 L·atm/mol·K.
- Plug into the Formula: Substitute the values of M, P, R, and T into the equation: d = MP/RT.
- Calculate Density (d): Calculate the density, which will be in g/L.
Example
Let’s say you want to calculate the density of nitrogen gas (N2) at a partial pressure of 0.5 atm and a temperature of 298 K.
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Molar Mass of N2: The molar mass (M) of N2 is approximately 28 g/mol.
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Partial Pressure (P): The partial pressure is 0.5 atm.
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Temperature (T): The temperature is 298 K.
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Ideal Gas Constant (R): R = 0.0821 L·atm/mol·K
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Calculation:
d = (28 g/mol 0.5 atm) / (0.0821 L·atm/mol·K 298 K)
d ≈ 0.572 g/L
Therefore, the density of nitrogen gas under these conditions is approximately 0.572 g/L.
Key Insights:
- Molar Mass Influence: The equation indicates that gas density increases with molar mass. This is because more mass is packed into the same volume with heavier molecules.
- Temperature Impact: Temperature has an inverse relationship with density. As temperature increases, density decreases due to the expansion of gas.
- Pressure Effect: Density is directly proportional to the pressure of the gas. Increasing the pressure squeezes more gas into the same volume, increasing the density.
- Partial Pressure Specificity: This equation only applies to the partial pressure of the gas you are trying to measure the density of. It does not apply to the total pressure of a gas mixture.
- The equation demonstrates the direct relationship between the gas density and its molar mass at standard temperature and pressure.
