An insulated container of gas has two chambers separated by an insulating partition. One of the chambers has volume $V_{1}$ and contains ideal gas at pressure $P_{1}$ and temperature $T_{1}$ . The other chamber has volume $V_{2}$ and contains ideal gas at pressure $P_{2}$ and temperature $T_{2}$ . If the partition is removed without doing any work on the gas, the final equilibrium temperature of the gas in the container will be

$\frac{T_{1}T_{2}(P_{1}V_{1}+P_{2}V_{2})}{P_{1}V_{1}T_{1}+P_{2}V_{2}T_{2}}$

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$\frac{T_{1}T_{2}(P_{1}V_{1}+P_{2}V_{2})}{P_{1}V_{1}T_{2}+P_{2}V_{2}T_{1}}$

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$\frac{P_{1}V_{1}T_{1}+P_{2}V_{2}T_{2}}{P_{1}V_{1}+P_{2}V_{2}}$

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$\frac{P_{1}V_{1}T_{2}+P_{2}V_{2}T_{1}}{P_{1}V_{1}+P_{2}V_{2}}$

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A gas mixture consists of 2 moles of O₂ and 4 moles of Ar at temperature T. Neglecting all vibrational modes, the total internal energy of the system is

4 RT

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15 RT

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9 RT

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11 RT

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Two thermally insulated vessels 1 and 2 are filled with air at temperatures $(T_{1},T_{2}),$ volume $(V_{1},V_{2}),$ and pressure $(P_{1},P_{2}),$ respectively. If the valve joining the two vessels is opened, the temperature inside the vessel at equilibrium will be

$T_{1}+T_{2}\; \;$

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$(T_{1}+T_{2})/2\; \;$

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$\frac{T_{1}T_{2}(P_{1}V_{1}+P_{2}V_{2})}{P_{1}V_{1}T_{2}+P_{2}V_{2}T_{1}}\; \;$

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$\; \frac{T_{1}T_{2}(P_{1}V_{1}+P_{2}V_{2})}{P_{1}V_{1}T_{1}+P_{2}V_{2}T_{2}}$

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Concepts Covered - 1

Heat, Internal energy and Work in Thermodynamics

Quantities Involved in First Law of Thermodynamics -

(a) Heat (Q) : It is the energy that is transferred between a system and its environment because of the temperature gradient.

(b) Work (W) : Work can be defined as the energy that is transferred from one body to the other owing to a force that acts between them.

(C) Internal energy (U) : Internal energy of a system is the energy possessed by the system due to molecular motion and molecular configuration.

Types of internal energy -

Due to molecular motion internal energy is kinetic internal energy (UK).
Due to molecular configuration, it is called internal potential energy (UP).

Important points :

Heat and work are path-dependent quantities and Internal energy is point function.
$$
\Delta W=P \Delta V=P\left(V_f-V_i\right)
$$

$\Delta W=$ positive if $V_f>V_i$ i.e. system expands against some external force.
$\Delta W=$ negative if $V_f<V_i$ i.e. system contracts because of some external force exerted by the surrounding.
The area of the P-v diagram on the volume axis gives the work done in a reversible process. Also for quasistatic process work is given by

$W=\int_{V_1}^{V_2} P . d V$

For a cyclic process the clockwise area will show positive work and the anticlockwise area will show negative work done.
The internal energy of an ideal gas is totally kinetic and is given by:

$U=\frac{3}{2} \cdot n \cdot R \cdot T$

So, the Internal energy of an ideal gas is the function of temperature only.

6. For heat transfer -

$\Delta Q=m L_{\text {(for change of state) }}$
$\Delta Q=m s \Delta T_{\text {(for change in temperature) }}$
or,

$$
\Delta Q=n c \Delta T
$$

Where, $c=$ molar specific heat capacity
Sign of dQ(Heat)
$d Q>0$ if heat is given to the system
$d Q<0$ if heat is extracted from the system