Question

In an isochoric process if ${T_1} = {27^ \circ }C$ and ${T_2} = {127^ \circ }C$ , then $\dfrac{{{P_1}}}{{{P_2}}}$ ​ will be equal to
A. $\dfrac{9}{{59}}$
B. $\dfrac{2}{3}$
C. $\dfrac{3}{4}$
D. None of these

Answer 1

Hint: We know that the parameters such as pressure and temperature vary with the given conditions of the system and surroundings but in this problem, volume remains constant for an isochoric process hence, use the Gay-Lussac Law which works at constant volume to calculate the ratio of $\dfrac{{{P_1}}}{{{P_2}}}$ for the given situation.

Complete Step by Step Solution:
Isochoric process in thermodynamics is a process during which the volume of a system remains constant that’s why it is also referred to as a constant-volume process. $\Delta V = 0$
It is given that the initial temperature is ${T_1} = {27^ \circ }C = 300K$
$\left( {^ \circ C + 273 = K} \right)$
And the Final temperature is ${T_2} = {127^ \circ }C = 400K$

Now, we know that at constant volume, the pressure of a gas is directly proportional to its absolute temperature (according to Gay-Lussac Law).
i.e., In an Isochoric Process, $\dfrac{P}{T} = $constant
or, $\dfrac{{{P_1}}}{{{P_2}}} = \dfrac{{{T_1}}}{{{T_2}}}$ … (1)
Substitute the values given in the question in eq. (1), we get
$ \Rightarrow \dfrac{{{P_1}}}{{{P_2}}} = \dfrac{{300}}{{400}}$
$ \Rightarrow \dfrac{{{P_1}}}{{{P_2}}} = \dfrac{3}{4}$
Thus, the ratio of $\dfrac{{{P_1}}}{{{P_2}}}$, according to the given situation is $\dfrac{3}{4}$.
Hence, the correct option is (C) $\dfrac{3}{4}$.

Note: In this conceptual-based problem, to determine the required ratio of $\dfrac{{{P_1}}}{{{P_2}}}$, use Gay-Lussac’s law, and hence insert the values of the two temperatures in the expression of the law to provide an appropriate solution. Always remember to use the mathematical proven relations to get the solution while composing an answer to this kind of conceptual problem.