AP Physics 2 Exam Question

An ideal gas undergoes a thermodynamic process shown in the P-V diagram below:

1. What type of process does the gas go through from Point A to Point B? Explain your answer.

Assume the ideal gas obeys the ideal gas law, PV = nRT, where P is pressure, V is volume, n is the number of moles of gas, R is the ideal gas constant, and T is temperature.

Answer:

The gas goes through an isothermal process from Point A to Point B.

Explanation: In an isothermal process, the temperature of the gas remains constant. From the P-V diagram, we can see that the curve connecting Point A to Point B is a hyperbola. In an isothermal process, according to the ideal gas law, the product of pressure and volume remains constant. Therefore, the curve is an isotherm.

2. Calculate the work done by the gas when it undergoes the process from Point B to Point C.

Answer:

To calculate the work done by the gas, we need to find the area under the curve connecting Point B to Point C on the P-V diagram.

The area under a curve on a P-V diagram represents the work done by the gas during that process.

To find the area under the curve, we can divide it into two regions: the rectangular region and the triangular region.

Let's calculate the work done for each region separately:

1. Rectangular Region: The length of the rectangle is the change in volume ($ΔV$) from Point B to Point C, which is given as $ΔV = V_C - V_B$. The width of the rectangle is the constant pressure ($P_B$) during the process from Point B to Point C. Therefore, the work done in the rectangular region is given by: [Work_{\text{rectangle}} = P_B \cdot ΔV]

2. Triangular Region: The base of the triangle is the change in volume ($ΔV$) from Point C to Point D, which is given as $ΔV = V_D - V_C$. The height of the triangle can be calculated using the equation of the isotherm curve, which is $P = \frac{nRT}{V}$. Therefore, the height of the triangle at Point C is given by $P_C = \frac{nRT}{V_C}$. The work done in the triangular region is given by: [Work_{\text{triangle}} = \frac{1}{2} \cdot P_C \cdot ΔV]

The total work done by the gas from Point B to Point C is the sum of the work done in the rectangular region and the work done in the triangular region: [Work_{\text{total}} = Work_{\text{rectangle}} + Work_{\text{triangle}}]

Substituting the values we have: [Work_{\text{total}} = P_B \cdot ΔV + \frac{1}{2} \cdot P_C \cdot ΔV]

The change in volume from Point B to Point C is the difference in volumes ($V$) at those points, which is given as: [ΔV = V_C - V_B]

Therefore, the final expression for the total work done by the gas is: [Work_{\text{total}} = P_B \cdot (V_C - V_B) + \frac{1}{2} \cdot P_C \cdot (V_D - V_C)]

Note: The given P-V diagram does not provide specific values for pressure or volume, so the final expression cannot be simplified further without numerical values.

This is how you calculate the work done by the gas when it undergoes the process from Point B to Point C.