# Heat engine.

#### j88k

1. The problem statement, all variables and given/known data

A heat engine using 5.00 g of helium gas is initially at STP. The gas goes through the following closed cycle:
- Isothermal compression until the volume is halved.
- Isobaric expansion until the volume is restored to its initial value.
- Isochoric cooling until the pressure is restored to its initial value.

a)How much work does this engine do per cycle?
b)What is its thermal efficiency?

2. Relevant equations

thermal efficiency = Wout/QH

(thermal efficiency)Carnot = 1 - TC/TH

3. The attempt at a solution

for part a) I got:

isothermal work = nRTln(V at 2/V at 1) = -1967 J
isobaric work = p(V at 3 - V at 2) = 2836 J
isochoric work = 0

total work done in one cycle = -1967 + 2836 = 869 J

for pat b) i got:

from 1 to 2:
Q = -W = 1967 J

from 2 to 3:
Q = nC_p(deltaT), C_p for a monatomic gas is (5/2)R, this comes from the equipartition theorem
Q = (1.25)(5R/2)(273K) = 7094 J

from 3 to 1:
Q must be negative since there is a work output so therefore Q in this process is irrelevant since we only need the Q that is added to the system

efficiency = (total work done in one cycle)/(heat added in one cycle)
efficiency = (869 J)/(7094 J + 1967 J) = 0.096=9.6% But apparently thats the wrong answer!

I don't know what's wrong. Help!

Last edited:

#### alphysicist

Homework Helper
Hi j88k,

1. The problem statement, all variables and given/known data

A heat engine using 5.00 g of helium gas is initially at STP. The gas goes through the following closed cycle:
- Isothermal compression until the volume is halved.
- Isobaric expansion until the volume is restored to its initial value.
- Isochoric cooling until the pressure is restored to its initial value.

a)How much work does this engine do per cycle?
b)What is its thermal efficiency?

2. Relevant equations

thermal efficiency = Wout/QH

(thermal efficiency)Carnot = 1 - TC/TH

3. The attempt at a solution

for part a) I got:

isothermal work = nRTln(V at 2/V at 1) = -1967 J
isobaric work = p(V at 3 - V at 2) = 2836 J
isochoric work = 0

total work done in one cycle = -1967 + 2836 = 869 J

for pat b) i got:

from 1 to 2:
Q = -W = 1967 J
Looking at this part: If the gas is being compressed and the temperature is not changing, is heat entering or leaving the system? (In the efficiency formula you only count heat going into the system.) Does this help?

#### j88k

so the answer would be (869)/(7094)=0.122=12.2% ?

Staff Emeritus
Homework Helper
Looks good!

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