H2 Physics 13 - Thermodynamic Systems: First Law Applied to an Ideal Gas

Step 1 - State the problem

An ideal monatomic gas in a cylinder is heated at constant pressure from three hundred kelvin to five hundred kelvin.

Step 1 - State the problem

The gas contains zero point two five moles.

Step 1 - State the problem

We want to find the heat supplied, the work done by the gas, and the increase in internal energy.

Step 1 - State the problem

Take the molar gas constant R as eight point three one joules per mole per kelvin.

Step 2 - Find the work done at constant pressure

At constant pressure, the work done by the gas is W equals p delta V.

Step 2 - Find the work done at constant pressure

Using the ideal gas law, p V equals n R T, the change in p V at constant pressure is n R delta T.

Step 2 - Find the work done at constant pressure

So W equals n R delta T equals zero point two five times eight point three one times two hundred.

Step 3 - Find the change in internal energy

For a monatomic ideal gas, the internal energy depends only on temperature.

Step 3 - Find the change in internal energy

The change in internal energy delta U equals three halves n R delta T.

Step 3 - Find the change in internal energy

Substituting gives three halves times zero point two five times eight point three one times two hundred, which equals six hundred and twenty-three joules.

Step 4 - Apply the first law to find heat supplied

The first law of thermodynamics says the heat supplied Q equals the change in internal energy plus the work done by the gas.

Step 4 - Apply the first law to find heat supplied

Q equals delta U plus W equals six hundred and twenty-three plus four hundred and sixteen.

Step 4 - Apply the first law to find heat supplied

That gives Q approximately one thousand and thirty-nine joules.

Step 4 - Apply the first law to find heat supplied

Notice that only about forty percent of the heat goes into doing work. The rest raises the internal energy.

Step 5 - Verify using molar heat capacity at constant pressure

We can check using Q equals n C p delta T, where C p for a monatomic ideal gas is five halves R.

Step 5 - Verify using molar heat capacity at constant pressure

C p equals five halves times eight point three one equals twenty point eight joules per mole per kelvin.

Step 5 - Verify using molar heat capacity at constant pressure

Q equals zero point two five times twenty point eight times two hundred, which gives one thousand and thirty-nine joules, confirming our answer.

Step 5 - Verify using molar heat capacity at constant pressure

Remember: C p is always greater than C v because extra heat is needed to do expansion work at constant pressure.