Problem 112 - Chemistry: An Atoms-Focused Approach

Question

Under what conditions is the pressure exerted by a real gas less than that predicted for an ideal gas?

Step-by-Step Solution

Answer

Answer: The pressure exerted by a real gas is less than the pressure exerted by an ideal gas under the following conditions: 1) The ratio of the number of moles squared (n^2) multiplied by the attractive forces parameter (a) to the volume squared (V^2) is large, and 2) the ratio of the number of moles (n) multiplied by the temperature (T) and finite size parameter (b) to the product of the volume (V) and the quantity (V-nb) is small. These conditions generally occur at low temperatures and high pressures, as the volume is low, and the attractive forces between molecules have a more significant impact on the overall pressure exerted by the real gas.

Step 1: Ideal Gas Equation

Recall the ideal gas equation, which is given by: PV = nRT where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature in Kelvin. For an ideal gas, the pressure is given by: P_ideal = (nRT) / V

Step 2: Van der Waals Equation

The van der Waals equation is a more accurate equation for real gases. It is given by: (P + (n^2a/V^2))(V-nb) = nRT Where P is pressure, V is the volume, n is the number of moles, R is the gas constant, T is temperature, a is a parameter that accounts for the attractive forces between molecules, and b is a parameter related to the finite size of the molecules. The pressure exerted by a real gas is given by: P_real = ((nRT) / (V-nb)) - (n^2 * a/V^2) Under what conditions is the pressure exerted by a real gas less than the pressure exerted by an ideal gas? This happens when: P_real < P_ideal

Step 3: Compare the ideal and real gas equations

Now we will substitute the expressions of P_ideal and P_real from Step 1 and Step 2 into the inequality and simplify. ((nRT) / (V-nb)) - (n^2 * a/V^2) < (nRT) / V

Step 4: Solve the inequality

To solve the inequality, let's first subtract (nRT) / (V-nb) from both sides: - (n^2 * a/V^2) < (nRT) / V - (nRT) / (V-nb) Now we will combine the terms on the right-hand side by finding a common denominator: - (n^2 * a/V^2) < [nRT(V-nb) - nRT(V)] / (V(V-nb)) Simplify the numerator: - (n^2 * a/V^2) < -nRTnb / (V(V-nb)) Multiply both sides of the inequality by -1 and recall that multiplying an inequality by a negative number reverses the direction of the inequality: (n^2 * a/V^2) > nRTnb / (V(V-nb))

Step 5: Determine the conditions under which the inequality is true

Analyzing the inequality derived in Step 4, pressure exerted by a real gas (P_real) is less than the pressure exerted by an ideal gas (P_ideal) under the following conditions: 1. The ratio of the number of moles squared (n^2) multiplied by the attractive forces parameter (a) to the volume squared (V^2) is large. 2. The ratio of the number of moles (n) multiplied by the temperature (T) and finite size parameter (b) to the product of the volume (V) and the quantity (V-nb) is small. These conditions generally occur at low temperatures and high pressures, as the volume is low, and the attractive forces between molecules have a more significant impact on the overall pressure exerted by the real gas.