Question:

A sample of gas undergoes a thermodynamic process as shown in the diagram below.

The gas initially has a volume of 2.0 L and a pressure of 3.0 atm. During the process, the gas expands isothermally to a final volume of 4.0 L. Determine the final pressure of the gas, assuming ideal gas behavior.

(A) 1.5 atm

(B) 2.0 atm

(C) 3.0 atm

(D) 4.0 atm

Answer:

To determine the final pressure of the gas after the isothermal expansion, we can use the ideal gas law, which states:

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

Since the process is isothermal, the temperature of the gas remains constant. Therefore, $\Delta T = 0$.

The ideal gas law can be rearranged to solve for pressure:

Given that the volume initially is 2.0 L and the final volume is 4.0 L, we can determine the ratio of the initial and final volumes:

$\frac{{V_f}}{{V_i}} = \frac{{4.0 \, \text{L}}}{{2.0 \, \text{L}}} = 2$

Since the temperature remains constant, n and R remain constant as well. Therefore, the ratio of the initial and final pressures will be the same as the ratio of the initial and final volumes:

$\frac{{P_f}}{{P_i}} = \frac{{V_i}}{{V_f}} = \frac{1}{2}$

Now, we can substitute the given initial pressure of 3.0 atm into the equation:

$\frac{{P_f}}{{3.0 \, \text{atm}}} = \frac{1}{2}$

Multiply both sides of the equation by 3.0 atm:

$P_f = \frac{1}{2} \times 3.0 \, \text{atm}$

$P_f = 1.5 \, \text{atm}$

Therefore, the final pressure of the gas is 1.5 atm.

The correct answer is (A) 1.5 atm.