Assume that you have 15.00 mol of N2 in a volume of 0.600 L at 300 K. Calculate the pressure in atmospheres using both the ideal gas law and the van der Waals equation. For N2, a = 1.351 L^2 atm/mol^2 and b = 0.0387 L/mol.

Verified step by step guidance

1. Step 1: Start with the Ideal Gas Law, which is given by the equation: \( PV = nRT \). Here, \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the ideal gas constant (0.0821 L atm/mol K), and \( T \) is the temperature in Kelvin.
2. Step 2: Substitute the given values into the Ideal Gas Law equation: \( P = \frac{nRT}{V} \). Use \( n = 15.00 \) mol, \( R = 0.0821 \) L atm/mol K, \( T = 300 \) K, and \( V = 0.600 \) L.
3. Step 3: Calculate the pressure using the Ideal Gas Law by plugging in the values from Step 2 into the equation \( P = \frac{nRT}{V} \).
4. Step 4: Now, use the van der Waals equation, which is: \[ \left( P + \frac{a(n/V)^2}{n^2} \right)(V - nb) = nRT \]. Here, \( a \) and \( b \) are van der Waals constants specific to the gas, and \( n \), \( V \), and \( T \) are as defined previously.
5. Step 5: Substitute the given values and constants into the van der Waals equation: \( a = 1.351 \) L^2 atm/mol^2, \( b = 0.0387 \) L/mol, \( n = 15.00 \) mol, \( V = 0.600 \) L, and \( T = 300 \) K. Solve for \( P \) by rearranging the equation and performing the necessary calculations.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry that relates the pressure (P), volume (V), number of moles (n), and temperature (T) of an ideal gas through the equation PV = nRT. Here, R is the ideal gas constant. This law assumes that gas particles do not interact and occupy no volume, making it a good approximation under many conditions, but it may not hold true at high pressures or low temperatures.

Van der Waals Equation

The Van der Waals equation is an adjustment of the Ideal Gas Law that accounts for the volume occupied by gas molecules and the attractive forces between them. It is expressed as (P + a(n/V)^2)(V - nb) = nRT, where 'a' and 'b' are constants specific to each gas. This equation provides a more accurate description of real gas behavior, especially under conditions where deviations from ideality occur.

Real Gases vs. Ideal Gases

Real gases deviate from ideal behavior due to intermolecular forces and the finite volume of gas particles. While the Ideal Gas Law assumes no interactions and negligible volume, real gases exhibit attractions and repulsions that can affect pressure and volume, particularly at high pressures and low temperatures. Understanding these differences is crucial for accurately predicting gas behavior in various conditions.