# When a quantity of monatomic ideal gas expands at a constant pressure of 4.00 x 10^{4} Pa, the...

## Question:

When a quantity of monatomic ideal gas expands at a constant pressure of {eq}\rm 4.00 \times 10^{4} \ Pa {/eq}, the volume of the gas increases from {eq}\rm 2.00 \times 10^{-3} \ m^3 {/eq} to {eq}\rm 8.44 \times 10^{-3} \ m^3 {/eq}. What is the change in the internal energy of the gas?

## Internal Energy of an Ideal Gas:

Internal energy is a thermodynamic parameter defined as a sum of kinetic energies of all molecules of the sample and total potential energy of all inter-molecular interactions. For an ideal gas, we can neglect the potential energy. The average kinetic energy of gas molecules could be estimated in the frameworks of the kinetic theory where it was demonstrated that the average kinetic energy of the molecules is related with gas temperature.

## Answer and Explanation:

The change in the internal energy of the gas {eq}\Delta U=386\ \rm J {/eq}.

The internal energy of a mono-atomic ideal gas

{eq}U=3/2nRT {/eq}

where *n* is the number of moles, *R* is the gas constant, *T* is the temperature.

On the other hand the equation of state of an ideal gas

{eq}pV=nRT {/eq}

Combining the two equations we obtain

{eq}U=3/2pV {/eq}

The change of the internal energy {eq}\Delta U = 3/2p\Delta V = 3/2 \cdot \rm 4.00 \times 10^{4} (\rm 8.44 \times 10^{-3}-\rm 2.00 \times 10^{-3})= 386 \ J {/eq}

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