# How do temperature differences affect the work done in an ideal Carnot process?

• Pouyan
In summary, the problem involves finding the work necessary to cool hydrogen gas from 0 C to -10 C using an ideal Carnot process. The equations used include Qc/W = Tc/Th -Tc and W= Qc (Th-Tc)/Tc. The solution involves using integrals when the temperature of one of the reservoirs changes.
Pouyan

## Homework Statement

An ideal Carnot process operates between a heating bath with the temperature of 20 C and a mole of hydrogen is in a container with constant volume. During the process, the work is done to remove heat from the hydrogen gas and emit heat to the heat bath. Calculate the work necessary to cool the hydrogen gas to temperature -10 C, if the original's temperature is 0 C.

## Homework Equations

With heat pump and Carnot:
Qc/W = Tc/Th -Tc

and in this case W= Qc (Th-Tc)/Tc

Qc = ν*2.5*R*dTc

## The Attempt at a Solution

Well I figured I can solve Qc = 1 * 2.5* 8.31 * (10) J = 207.75 J
and W = 207.75 (30)/263 J= 23.7 J
But I see in my solution:

Qc =ν*2.5*R*dTc and W= Qc (Th-Tc)/Tc ⇒ dW =ν*2.5*R*((Th-Tc)/Tc)* dTc

and W = ∫dW=∫ν*2.5*R*((Th-Tc)/Tc)* dTc and this is between 273 K and 263 K. And W = 19.4 J.

I know with these engines we will do work with temperature differences (Th-Tc) but should we always use integrals if Th or Tc is changing like this exercise ?!
Think if for example we would find the value of work to change the heat reservoir from 20 C to 30 C, should we use again integrals to do this ?!

Yes, if the temperature of one of the reservoirs changes, you must integrate.

Chestermiller

## 1. What is the definition of work in thermodynamics?

In thermodynamics, work is defined as the energy transferred to or from a system due to a change in its volume or pressure.

## 2. How is work related to the Carnot process?

The Carnot process is a theoretical thermodynamic cycle that describes the most efficient way to convert heat into work. It consists of four reversible processes: isothermal expansion, adiabatic expansion, isothermal compression, and adiabatic compression. Work is done during the isothermal expansion and compression stages, as the system expands or compresses against a constant external pressure.

## 3. What is the formula for calculating work in the Carnot cycle?

The formula for calculating work in the Carnot cycle is W = Qh - Qc, where W is the work done, Qh is the heat absorbed from the hot reservoir, and Qc is the heat rejected to the cold reservoir.

## 4. What is the efficiency of the Carnot process?

The efficiency of the Carnot process is given by the formula η = 1 - Tc/Th, where Tc is the temperature of the cold reservoir and Th is the temperature of the hot reservoir. This means that the efficiency of the Carnot cycle is dependent on the temperatures of the reservoirs, and it is always less than 100%.

## 5. How is the Carnot cycle used in practical applications?

The Carnot cycle is used as a theoretical benchmark for the maximum efficiency of real heat engines. It helps engineers in designing more efficient engines by identifying areas for improvement. It is also used in the development of refrigerators and heat pumps, as well as in the study of thermodynamic processes and principles.

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