Question 94: Thermodynamics & KTG

Question

If 2 mole of an ideal monoatomic gas at temperature $T$, is mixed with 6 mole of another ideal monoatomic gas at temperature $2 T$ then the temperature of mixture is:

Accepted Answer

The internal energy $(U)$ of $n$ moles of an ideal gas at temperature $T$ is given by $U=n C_v T$, where $C_v$ is the molar specific heat at constant volume.
For the mixture, total initial internal energy $\mathrm{U}_{	ext {initial }}=\mathrm{U}_1+\mathrm{U}_2$
Total final internal energy $\mathrm{U}_{	ext {final }}=\left(\mathrm{n}_1+\mathrm{n}_2ight) \mathrm{C}_{\mathrm{v}, 	ext { mix }} \mathrm{T}_{	ext {mix }}$
Since both gases are monoatomic, their molar specific heats are the same,
$$ C_{v 1}=C_{v 2}=C_{v, \operatorname{mix}}=\frac{3}{2} R $$
Therefore, using conservation of energy (total initial internal energy is equal to final internal energy).
$$ \mathrm{n}_1 \mathrm{C}_{\mathrm{v}} \mathrm{~T}_1+\mathrm{n}_2 \mathrm{C}_{\mathrm{v}} \mathrm{~T}_2=\left(\mathrm{n}_1+\mathrm{n}_2ight) \mathrm{C}_{\mathrm{v}} \mathrm{~T}_{\operatorname{mix}} $$
$$\Rightarrow $$ $$ \mathrm{T}_{\mathrm{mix}}=\frac{\mathrm{n}_1 \mathrm{~T}_1+\mathrm{n}_2 \mathrm{~T}_2}{\left(\mathrm{n}_1+\mathrm{n}_2ight)} $$
Moles of first gas is $\mathrm{n}_1=2$
Temperature of first gas is $\mathrm{T}_1=\mathrm{T}$
Moles of second gas $\mathrm{n}_2=6$
Temperature of second gas $\mathrm{T}_2=2 \mathrm{~T}$
On substituting the values into expression for the temperature of mixture,
$$ \mathrm{T}_{\mathrm{mix}}=\frac{(2 	imes \mathrm{T})+(6 	imes 2 \mathrm{~T})}{2+6} $$
$$\Rightarrow $$ $\mathrm{T}_{\operatorname{mix}}=\frac{2 \mathrm{~T}+12 \mathrm{~T}}{8}$
$$\Rightarrow $$ $\mathrm{T}_{\mathrm{mix}}=\frac{14 \mathrm{~T}}{8}$
$$\Rightarrow $$ $$ \mathrm{T}_{\mathrm{mix}}=\frac{7}{4} \mathrm{~T} $$
Therefore, the correct option is (D).

Answer

$\frac{5}{2} T$

Answer

$\frac{5}{4} T$

Answer

$\frac{7}{2} T$

Answer

$\frac{7}{4} T$

Thermodynamics & KTG

If 2 mole of an ideal monoatomic gas at temperature T, is mixed with 6 mole of another ideal monoatomic gas at temperature 2 T then the temperature of mixture is:

Choose the correct answer: