The Equation of State
The Equation of State
- A gas has four measurable properties: pressure, volume, temperature and amount.
- One equation links all four. It is exact only for a gas called an ideal gas, so we say what that means first.
What an ideal gas is
- An is a gas for which pressure multiplied by volume is proportional to thermodynamic temperature.
- Both parts of that sentence score. The proportionality is one mark, and naming the thermodynamic temperature is the second (9702/41/M/J/24 Q3(a)(i)).
- The amount of gas must be fixed. If gas leaks out, the relation no longer holds.
- Real gases follow this closely at everyday pressures and temperatures.
- A real gas stops following it closely at high density or cooled close to the temperature at which it turns into a liquid. The molecules are then close together, so the space they take up and the forces between them are no longer small.
Definition
2 marksWhat is meant by ideal gas?
Model answer: An ideal gas is a gas for which pressure multiplied by volume is proportional to thermodynamic temperature, for a fixed amount of gas.
The equation of state
- At constant temperature, pressure is inversely proportional to volume for a fixed amount of gas.
- At constant pressure, volume is proportional to thermodynamic temperature.
- Put the two relationships together. The constant depends on the amount of gas, so it is written using and .
Symbols
- = pressure of the gas (Pa)
- = volume of the gas (m³)
- = amount of gas (mol)
- = molar gas constant, 8.31 (J K⁻¹ mol⁻¹)
- = thermodynamic temperature (K)
- When it applies. The gas is ideal, and the amount of gas is fixed.
- Units are not optional. must be in kelvin, in m³ and in Pa. A temperature in °C put straight into this equation gives a wrong answer every time.
- Two states of the same gas. If the same fixed amount of gas changes from state 1 to state 2, use the relation below when the question does not give .
Worked example
Exam version: find the temperature of a gas
A cylinder has volume 0.0260 m³ and contains 0.740 mol of an ideal gas at a pressure of Pa. Show that the temperature of the gas is 234 °C. (9702/42/M/J/24 Q2(b)(i))
- Rearrange the equation of state for .
- Substitute the values in SI units.
- The question asks for °C, so convert at the end.
Answer
Work in kelvin inside the equation. Change to °C only in the final line, and only if the question asks for it.
Your turn9702-style
A fixed amount of gas is at Pa, m³ and 27 °C. Later it is at Pa, m³ and 87 °C. Show that the readings agree with the ideal-gas relationship.
Show worked answer
- Change both temperatures to kelvin: 300 K and 360 K.
- Calculate the state-1 value.
- Calculate the state-2 value.
Answer: the two values are equal, so the readings agree with the ideal-gas relationship.
Common mistake
Two common errors are a temperature left in °C, and a volume left in cm³ or dm³. Change both before the first line of working. Use the conversion below.