Volumes at C and D in a Carnot cycle

[Qwurty2.0]

Homework Statement

A Carnot cycle, shown in Fig. 20-7, has the following conditions: V_a = 7.5 L, V_b = 15 L, T_H = 470°C, and T_L = 260°C. The gas used in the cycle is 0.50 mil of a diatomic gas, y = 1.4. Calculate (a) the pressures at a and b; (b) the volumes at c and d. (The rest of the problem...)

Homework Equations

PV^y = Constant

The Attempt at a Solution

I already solved part (a), with P_a = 4.1 x 10⁵ Pa and P_b = 2.1 x 10⁵ Pa.

For part (b), I tried
PV^y = constant
nRT⋅V^y-1 = constant.

Annnnd then I get lost. I don't understand what constant represents or if I am even on the right track. The book says V_c = 34 L and V_d = 17 L.

 

[vela]

What do the points a, b, c, and d represent in the Carnot cycle? You didn't post the figure.

 

[Qwurty2.0]

My apologies, here is the figure:

Points a, b, c, and d represent stages of the Carnot engine as it works.

 

[vela]

For an adiabatic process, you have $PV^\gamma = \text{constant}$. Since points b and c are on the same adiabat, you can say that $P_b V_b^\gamma = P_c V_c^\gamma$.

If it still seems confusing, consider the analogous situation for an isothermal process. The righthand side of the ideal gas law, $PV = nRT$, is constant for an isothermal process. In other words, on an isotherm, we have that $PV = \text{constant}$. Points a and b are on the same isotherm, so you can say $P_a V_a = P_b V_b$, which should look very familiar to you.