## Mean Kinetic Energy of Helium

Determine the change of mean kinetic energy of helium molecules with the mass of 100 g provided that during an isochoric process we supply it with the heat of 3.5 kJ.

Consider helium to be an ideal gas.

• #### Hint 1

Remember that during an isochoric process the volume of the gas does not change and the system therefore does not perform work. It therefore means that the supplied heat Q is all spent on increasing the internal energy U of the gas.

• #### Hint 2

Internal energy U of ideal gas is given only by the kinetic energy Ek of its molecule motion.

• #### Hint 3

Imagine that all supplied heat is divided evenly among all molecules of the gas. This is why the unknown change of mean kinetic energy $$\mathrm{\Delta}\bar{E}_{k}$$ of the molecules is given by the ratio of supplied heat Q and the total number N of molecules in the given amount of gas:

$\mathrm{\Delta}\bar{E}_{k}=\frac {Q}{N}.$
• #### Hint 4

To determine the number of molecules N in the given amount of gas using known variables, use relations between the number of particles, amount of substance n, molar mass Mm and the mass of the gas m.

• #### Given Values

 m = 100 g = 0.1 kg mass of helium molecules Q = 3.5 kJ = 3500 J heat supplied during an isochoric process $$\mathrm{\Delta}\bar{E}_k$$ = ? change of mean energy of helium molecules

Table values:

 Mm = 4 g mol−1 = 0.004 kg mol−1 molar mass of helium NA = 6.023·1023 mol−1 Avogadro's constant
• #### Analysis

During an isochoric process the volume of the gas does not change so the system does not perform work. This according to the First Law of Thermodynamics means that all supplied heat is spent on increasing internal energy of the gas. In the case of an ideal gas, it is given only by the kinetic energy of the gas molecules.

If we talk about the change of mean kinetic energy of the molecules we can imagine that the whole supplied heat is evenly divided among all molecules of the gass. This is why the unknown change is given by the ratio of supplied heat and the total number of molecules in the given amount of gas.

Using basic relations between molar mass, substance mass, amount of substance and number of particles we finally obtain the resulting formula for unknown mean kinetic energy of the molecules.

• #### Solution

During an isochoric process the volume of the gas does not change so the system does not perform work. According to the First Law of Termodynamics it is true that the whole supplied heat Q is spent on increasing the internal energy U of the gas, i.e.

$\mathrm{\Delta}U=Q.$

In the case of ideal gas, the internal energy is given only by the kinetic energy Ek of the molecule motion.

We have to determine mean kinetic energy $$\mathrm{\Delta}\bar{E}_k$$ of the molecules. Imagine that all the supplied heat is evenly divided among all of the molecules of the gas and therefore the unknown change is given by the ratio of supplied heat Q and the total number N of molecules in the given amount of gas. Therefore the following relationship applies

$\mathrm{\Delta}\bar{E}_{k}=\frac {Q}{N}.$

However we do not know the number of molecules. We need to express it using known variables. We express it as the product of amount of substance n of the gas and Avogadro' constant NA. Therefore it is true that

$\mathrm{\Delta}\bar{E}_{k}=\frac {Q}{N}=\frac{Q}{nN_A}.$

The amount of substance n of the gas is given by

$n=\frac{m}{M_m},$

where m is the mass of the gas and Mm its molar mass.

After substitution and adjustment we obtain the change of mean kinetic energy of the molecules we were looking for

$\mathrm{\Delta}\bar{E}_{k}=\frac {QM_m}{mN_A}.$
• #### Numerical Solution

$\mathrm{\Delta}\bar{E}_{k}=\frac {QM_m}{mN_A}=\frac {3500\cdot{0{,}004}}{0.1\cdot{6.023}\cdot{10^{23}}}\,\mathrm{J}\dot{=}2.3\cdot{10^{-22}}\,\mathrm{J}$   