# Carnot cycle

1. Mar 29, 2013

### Abigale

Hallo,

A->B: isothermal
C->D: isothermal

I want to understand how to calculate the work done in the isothermal processes.

$W_{AB}= T_{H}Nk\ln{(\frac{V_{A}}{V_{\text{B}}})}\\ W_{CD}= -T_{n}Nk\ln{(\frac{V_{A}}{V_{\text{B}}})}$

But if i callculate i get:
$W_{CD}= -T_{n}Nk\ln{(\frac{V_{D}}{V_{C}})}$

So the question is, why is:
$(\frac{V_{A}}{V_{\text{B}}}) = (\frac{V_{D}}{V_{C}})$ ?

Thx
Abbigale

2. Mar 29, 2013

### Simon Bridge

Last edited: Mar 29, 2013
3. Mar 29, 2013

### Andrew Mason

Abigale,

From the first law we know that ΔU + W = Q where W is the work done BY the gas. But since ΔU = 0 in one complete cycle, W = Q = heat flow in from hot register - heat flow out to cold register = |Qh| - |Qc|. Heat flow occurs only from A-B and C-D, so the difference in these heat flows is the work done for the entire cycle. That means the work done in the adiabatic parts nets out to 0.

The efficiency is:

$$\eta = 1 - \frac{|Q_c|}{|Q_h|} = 1 - \frac{T_cNk\ln\left(\frac{V_D}{V_C}\right)}{T_hNk\ln\left(\frac{V_A}{V_B}\right)}$$

But we also know that for a Carnot cycle:

ΔS = Qh/Th + Qc/Th = 0 so |Qc/Qh| = Tc/Th

This necessarily means that $(\frac{V_{A}}{V_{\text{B}}}) = (\frac{V_{D}}{V_{C}})$

AM

4. Mar 30, 2013

### Abigale

Hey,
but is it possible to schow the Relation, just by looking at the pv-diagram?

5. Mar 30, 2013

### Andrew Mason

Yes, if the PV graph showed the scale.

AM

6. Mar 31, 2013

### Philip Wood

The equations you give for isothermal work are for an ideal gas.

Since we're dealing with an ideal gas we can use the adiabatic equations for an ideal gas, for BC and DA.

So $p_B V_B^{\gamma} = p_C V_C^{\gamma}$

and $p_A V_A^{\gamma} = p_D V_D^{\gamma}$

dividing: $\frac{p_B}{p_A} \frac{V_B^{\gamma}}{V_A^{\gamma}} = \frac{p_C}{p_D} \frac{V_C^{\gamma}} {V_D^{\gamma}}$

But for the isothermals: $\frac{p_B}{p_A} = \frac{V_A}{V_B}$ and $\frac{p_C}{p_D} = \frac{V_D}{V_C}.$

We can now eliminate the pressure ratios from the previous equation, giving...

$\frac{V_B^{\gamma - 1}}{V_A^{\gamma - 1}} = \frac{V_C^{\gamma - 1}} {V_D^{\gamma - 1}}$

So $\frac{V_B}{V_A} = \frac{V_C} {V_D}$

7. Mar 31, 2013

### Andrew Mason

It might be easier to use the adiabatic condition in terms of temperature and volume:

$TV^{\gamma - 1} = K$ from which it is apparent that adiabatic paths between the same two temperatures result in the same proportional changes in volume.

AM

8. Apr 1, 2013

### Philip Wood

Much neater – if you can remember TVγ-1 = K. [Though, I suppose TVanything = K would suffice!]

We've got to be careful of the logic here. OP's argument leads to $\left| \frac{Q_{AB}}{Q_{CD}} \right| = \frac{T_{AB}}{T_{CD}}$. The temperatures which appear in this equation are (in my interpretation of the above argument) ideal gas scale temperatures, effectively defined by pV = nRT. But the Second Law leads to a definition of thermodynamic temperature based on Carnot cycles for any working substance (not just an ideal gas) that is essentially $\frac{T_{AB}}{T_{CD}} = \left| \frac{Q_{AB}}{Q_{CD}} \right|$. Hence the OP's argument shows the equivalence of the thermodynamic scale and the ideal gas scale.

Last edited: Apr 1, 2013