Isothermal Work Calculator: Work Done by an Ideal Gas at Constant Temperature

The work done by an ideal gas expanding or being compressed at constant temperature.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Enter the amount of gas in moles, the constant temperature in kelvin, and either the starting and final volumes or the starting and final pressures. Choose the textbook gas constant 8.314 J/(mol K) or the exact value.

The calculator works out nRT, finds the pressures (or volumes) from the ideal gas law, and then the work done by the gas, W = nRT ln(V2/V1), which is the same as nRT ln(P1/P2) because pressure and volume are inversely proportional at constant temperature. It also shows the base-10 form, W = 2.303 nRT log10(V2/V1).

The temperature does not change, so the internal energy of an ideal gas does not change either, and the first law gives Q = W. An expansion gives positive work done by the gas and heat taken in. A compression gives negative work, which means work is done on the gas and heat leaves it.

Formulas used

  • W = nRT\ln\frac{V_2}{V_1} = nRT\ln\frac{P_1}{P_2}
  • W = 2.303\,nRT\log_{10}\frac{V_2}{V_1}
  • PV = nRT = \text{constant}
  • \Delta U = 0,\ Q = W

Worked example

With the default numbers in the calculator above:

  1. Amount of gas, temperature and gas constant: 4988.40 J
  2. Pressures from the ideal gas law (PV = nRT at constant T): 2494.20 kPa to 623.55 kPa
  3. Work done by the gas: 6915.39 J
  4. The same work using the base-10 logarithm: 6915.39 J
  5. Work in kilojoules: 6.92 kJ
  6. Change in internal energy and heat added (first law): dU = 0, Q = 6915.39 J

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: Gas constant R Exact value 8.314462618 J/(mol K).

  1. Amount of gas, temperature and gas constant: 4988.68 J
  2. Pressures from the ideal gas law (PV = nRT at constant T): 2494.34 kPa to 623.58 kPa
  3. Work done by the gas: 6915.78 J
  4. The same work using the base-10 logarithm: 6915.78 J
  5. Work in kilojoules: 6.92 kJ
  6. Change in internal energy and heat added (first law): dU = 0, Q = 6915.78 J

Common questions

What is an isothermal process?

A process in which the temperature of the gas stays constant. It has to be slow enough for heat to flow in or out so the gas stays at the temperature of its surroundings.

Why is the heat added equal to the work done?

For an ideal gas the internal energy depends only on temperature. If the temperature is constant the internal energy does not change, so by the first law all the heat added becomes work.

What is the work in an isothermal compression?

It is negative for the gas, because work is done on the gas. The same amount of heat has to leave the gas to keep the temperature constant.

Which gas constant should I use?

Use 8.314 J/(mol K) for textbook problems. The exact value is 8.314462618, and the difference is under 0.01 percent.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any physics or thermodynamics text (ideal gas, first law, isothermal process); Cengel and Boles, Thermodynamics: An Engineering Approach.

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