Question:
Two rigid boxes containing different ideal gases are placed on a table. Box $A$ contains one mole of nitrogen at temperature $T_0$, while box $B$ contains one mole of helium at temperature $(7/3)\, T_0$. The boxes are then put into thermal contact with each other, and heat flows between them until the gases reach a common final temperature (Ignore the heat capacity of boxes). Then, the final temperature of the gases, $T_f$, in terms of/$T_0$ is :
Two rigid boxes containing different ideal gases are placed on a table. Box $A$ contains one mole of nitrogen at temperature $T_0$, while box $B$ contains one mole of helium at temperature $(7/3)\, T_0$. The boxes are then put into thermal contact with each other, and heat flows between them until the gases reach a common final temperature (Ignore the heat capacity of boxes). Then, the final temperature of the gases, $T_f$, in terms of/$T_0$ is :
Updated On: Aug 1, 2022
- $T_{f} =\frac{3}{7}T_{0}$
- $T_{f} =\frac{7}{3}T_{0}$
- $T_{f} =\frac{3}{2}T_{0}$
- $T_{f} =\frac{5}{2}T_{0}$
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The Correct Option is C
Solution and Explanation
Here, change in internal energy of the system is ??cro. i.e., increase in internal energy of one is equal to decrease in internal energy of other.
$\Delta U_{A}=1\times\frac{5R}{2}\left(T_{f} -T_{0}\right)$
$\Delta U_{B}=1\times\frac{3R}{2}\left(T_{f} -\frac{7}{3}T_{0}\right)$
Now $\Delta U_{A}+\Delta U_{B}=0$
$\frac{5R}{2}\left(T_{f} -T_{0}\right)+\frac{3R}{2}\left(T_{f} -\frac{7T_{0}}{3}\right)=0$
$5T_{f} -5T_{0}+3T_{f} -7T_{0}=0$
$\Rightarrow 8T_{f} =12T_{0} \Rightarrow T_{f} =\frac{12}{8}T_{0}=\frac{3}{2}T_{0}$
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Concepts Used:
Internal Energy
Internal Energy is the microscopic energy contained within a system formed by the disordered movement of molecules (kinetic energy), Potential energy, and the nuclear energy present within the atoms of these molecules. It should be noted that the kinetic energy of molecules present in the system and not the kinetic energy of the system is calculated in the Internal Energy.
The Internal Energy is denoted by ‘U’ and is measured in Joules (J). This Internal Energy can increase with the increase in temperature and change of state or phase (from solid to liquid to gas). Heat Reservoirs store this Internal Energy.
Different Substances will have different Internal Energies depending on the atom, temperature, bonds, pressure, etc.
Forms of Internal Energy:
There are two forms of Internal Energy namely Kinetic Energy and Potential Energy
- Kinetic Energy: Kinetic Energy is the energy produced by the particles due to their motion. The motion of particles can be Rotational, Translational, Vibrational, etc., and hence the energies from these movements will be Translational Energies, Vibrational Energies, Rotational Energies, etc.
- Potential Energy: Potential Energy can be achieved from different types of interaction between particles, nuclear energy, electronic energy, etc.