Collection of Solved Problems in Physics

Change in the Internal Energy of Oxygen

Task number: 1286

• Hint 1

  What does the internal energy of ideal gas depend on?

• Solution of Hint 1

  The internal energy of an ideal gas depends only on its temperature. It is also a state function. Therefore the internal energy difference will for all three cases be the same.

• Hint 2

  Write the relation between the internal energy difference and the temperature difference.

• Expressing the formula for ΔU

  The internal energy difference ΔU is directly proportional to the temperature difference ΔT, the constant of proportionality is called the specific heat capacity c_V at constant volume. We can write the equation:

  \[\Delta U = c_V m \left(T_2 - T_1\right).\]

  The following equation applies for the molar heat capacity C_V:

  \[C_V=c_VM_m,\]

  where M_m is the molar mass. Therefore we can write:

  \[\Delta U=\frac{m}{M_m} C_V \left(T_2 - T_1\right).\]

  Now we substitute the temperature difference t₂ − t₁ for the difference of thermodynamic temperature T₂ − T₁. Thus we obtain the formula for the internal energy difference:

  \[\Delta U=\frac{m}{M_m} C_V \left(t_2 - t_1\right).\]

• Hint 3

  The molar heat capacity of oxygen at constant volume is

  \[C_V=\frac{5}{2}R.\]

• Analysis

  The internal energy of an ideal gas is a state function depending only on the temperature. The result will therefore be the same for all three cases.

  We will express the internal energy difference as a product of the amount of substance of oxygen, the molar heat capacity of oxygen at constant volume and the difference between the initial and final temperatures.

  The unknown amount of substance can be evaluated by dividing the mass by the molar mass of oxygen.

• Numerical values

  m = 100 g = 0.100 kg	the mass of oxygen
  t1 = 10 °C	the initial temperature of oxygen
  t2 = 60 °C	the final temperature of oxygen
  ΔU = ?	the internal energy difference of oxygen

  From The Handbook of Chemistry and Physics:

  R = 8.31 JK−1mol−1	the molar gas constant
  Mm = 32 g mol−1	the molar mass of oxygen O2

• Solution

  The internal energy of an ideal gas is a state function depending only on the temperature. This means that in all three considered cases, the result will be the same, because it depends only on the initial and the final temperature of the gas, and not on the way the gas was heated to the final temperature.

  The molar heat capacity of oxygen C_V at constant volume is

  \[C_V=\frac{5}{2}R\]

  and the amount of substance n is given by the formula

  \[n=\frac{m}{M_m},\]

  Therefore the internal energy difference ΔU can be expressed as follows:

  \[\Delta U=nC_V\left(T_2-T_1\right)= \frac{m}{M_m}\,\frac{5}{2}R\,\left(T_2-T_1\right).\]

  If we substitute the temperature difference t₂ − t₁ for the thermodynamic temperature difference T₂ − T₁ and adjust the fraction, we get

  \[\Delta U=\frac{5mR}{2M_m}\left(t_2-t_1\right).\]

• Numerical insertion

  \[\Delta U=\frac{5mR}{2M_m}\,\left(t_2-t_1\right)= \frac{5\cdot{ 0.100}\cdot{ 8.31}}{2\cdot{0.032}}\cdot \left(60-10\right)\,\mathrm{J}\dot{=}3250\,\mathrm{J}\]

• Answer

  The change in the internal energy in all considered cases is approximately 3250 J.