Question 721: Thermodynamics & KTG

Question

During an experiment, an ideal gas is found to obey an additional law $\mathrm{VP}^2=$ constant. The gas is initially at temperature ' T ' and volume ' V '. What will be the temperature of the gas when it expands to a volume 2 V ?

Accepted Answer

To solve this problem, we start by looking at the known relationships for an ideal gas. For an ideal gas, the equation is:
$ \mathrm{PV} = \mathrm{nRT} $
This equation can be rearranged to express pressure $ \mathrm{P} $ as:
$ \mathrm{P} = \frac{\mathrm{RT}}{\mathrm{V}} \quad 	ext{(Equation 1)} $
The problem states that the gas follows an additional relationship:
$ \mathrm{VP}^2 = 	ext{constant} $
Substituting Equation 1 into this relationship, we have:
$ \mathrm{V} \left(\frac{\mathrm{RT}}{\mathrm{V}}ight)^2 = 	ext{constant} $
This simplifies to:
$ \frac{\mathrm{R}^2\mathrm{T}^2}{\mathrm{V}} = 	ext{constant} $
From here, we isolate the temperature term:
$ \frac{\mathrm{T}^2}{\mathrm{V}} = 	ext{constant} $
Given two states of the gas, the first with volume $ \mathrm{V}_1 = \mathrm{V} $ and temperature $ \mathrm{T}_1 = \mathrm{T} $, and the second after expansion with volume $ \mathrm{V}_2 = 2\mathrm{V} $, we set up the relationship:
$ \left(\frac{\mathrm{T}_1}{\mathrm{T}_2}ight)^2 = \frac{\mathrm{V}_1}{\mathrm{V}_2} $
Substituting the given volumes:
$ \left(\frac{\mathrm{T}}{\mathrm{T}_2}ight)^2 = \frac{\mathrm{V}}{2\mathrm{V}} $
Solving for $\mathrm{T}_2$:
$ \left(\frac{\mathrm{T}}{\mathrm{T}_2}ight)^2 = \frac{1}{2} $
Taking the square root:
$ \frac{\mathrm{T}}{\mathrm{T}_2} = \frac{1}{\sqrt{2}} $
Thus, $\mathrm{T}_2$ is:
$ \mathrm{T}_2 = \sqrt{2} \mathrm{T} $
Therefore, when the gas expands to double its initial volume, its temperature will be $\sqrt{2} \mathrm{T}$.

Answer

$\sqrt{3} \mathrm{~T}$

Answer

$\sqrt{\frac{1}{2 T}}$

Answer

$\sqrt{2} \mathrm{~T}$

Answer

$\sqrt{3 \mathrm{~T}}$

Thermodynamics & KTG

During an experiment, an ideal gas is found to obey an additional law \mathrm{VP}^2= constant. The gas is initially at temperature ' T ' and volume ' V '. What will be the temperature of the gas when it expands to a volume 2 V ?

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