"The heat engine uses 1.00x10^-2 mol of a diatomic gas as the working substance. Determine the temperatures at the 3 points."
From 1-2, it's an adiabatic process and the constants are (P)(V^(7/5)) or (T)(V^(2/5)). From 2-3, it's an isobaric process with V/T as the constant. And from 3-1, it's an isochoric process with P/T as the constant.
I've found that P1 = 100,000Pa, P2 = 400,000Pa, and P3 = 400,000Pa and V2 = .001L.
I know that for diatomic atoms, you use Eth = d/2nRT, where d = 5.
But I'm absolutely stumped on how to get the missing V1, V3, T1, T2, and T3 variables.
Please help!
To determine the missing variables, we need to apply the appropriate equations and principles for each stage of the heat engine process.
Stage 1-2:
Since the process from 1-2 is adiabatic (Q = 0), we can use the adiabatic relationship for an ideal gas:
(P1)(V1^(γ)) = (P2)(V2^(γ))
Where γ is the heat capacity ratio or adiabatic index, which can be calculated as γ = (Cp) / (Cv). For a diatomic gas, Cp = 7/2R and Cv = 5/2R, so γ = 7/5.
We are given P1, P2, and V2, but we need to find V1. Rearranging the equation gives:
V1 = (P2 / P1) * (V2^(γ))
Substitute the known values to find V1.
Stage 2-3:
Since the process from 2-3 is isobaric (constant pressure), we have:
(V2 / T2) = (V3 / T3)
We know V2, and the pressure is constant (P2 = P3), so we need to find V3 and T3. Rearranging the equation gives:
V3 = (V2 / T2) * T3
Stage 3-1:
Since the process from 3-1 is isochoric (constant volume), we have:
(P3 / T3) = (P1 / T1)
We know P1 and P3 from your provided values, and the volume is constant (V1 = V3), so we need to find T1. Rearranging the equation gives:
T1 = (P1 / P3) * T3
By solving these equations successively, you can find the missing variables V1, V3, T1, T2, and T3.