# What is the relationship between the volume, pressure, and energy of gases? I know Boyle's and Charles' laws, but I can't figure out how to relate energy.

## No, no I don't understand what you're saying. Could you please explain?

Is there not a simple formula that relates volume, pressure, and energy? For example, if I had a balloon with hydrogen gas in it, how would I calculate the amount of energy stored by the gas?

## E = f/2 N k_b T

N is the number of gas molecules, k_b is Boltzmann's constant:

k_b = 1.38065*10^(-23) J/K

T is the absolute temperature and f is the number of degrees of freedom per molecule.

For a mono-atomic gas like e.g. helium f = 3. This is because an atom can move in three directions. Kinetic energy is 1/m v^2 = 1/2 m (v_x^2 + v_y^2 + v_z^2)

and all three components of the velocity will contribute equally to the internal energy.

In case of hydrogen, there are two additional degrees of freedom. The hydrogen molecule can rotate and you can choose two independent rotation axes orthogonal to the line connecting the two hydrogen atoms. The third possibility of the axis parallel to the molecule doesn't count at room temperatures, because the lowest possible energy above the ground state is so high comnpared to k_b T that this degree of freedom is "frozen". The same is true for the vibrational degrees of freedom at room temperature.

So for hydrogen at room temperature f = 5.

## The relationship between volume, pressure, and energy of gases is explained by the gas laws, specifically Boyle's law and Charles' law.

Boyle's law states that at a constant temperature, the volume of a gas is inversely proportional to its pressure. This can be mathematically represented as:
P1V1 = P2V2

Where P1 and V1 are the initial pressure and volume, and P2 and V2 are the final pressure and volume respectively.

Charles' law states that, at constant pressure, the volume of a gas is directly proportional to its temperature. This can be expressed mathematically as:
V1/T1 = V2/T2

Where V1 and T1 are the initial volume and temperature, and V2 and T2 are the final volume and temperature respectively.

To relate energy to volume, pressure, and temperature, we need to consider the concept of the ideal gas law. The ideal gas law is an equation that combines the relationships between pressure, volume, temperature, and the number of gas molecules.

The ideal gas law is given by the equation:
PV = nRT

Where P is the pressure, V is the volume, n is the number of gas molecules, R is the ideal gas constant, and T is the temperature (expressed in Kelvin).

The ideal gas law can be rearranged to solve for energy (E) by using the equation:
E = nRT

Here, E represents the energy of the gas, n is the number of gas molecules, R is the ideal gas constant, and T is the temperature (expressed in Kelvin).

Therefore, the relationship between the volume, pressure, and energy of gases can be understood by utilizing Boyle's law, Charles' law, and the ideal gas law.