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The ratio of two specific heats of gas $\dfrac{{{C_P}}}{{{C_V}}}$ for argon is 1.6 and for hydrogen is 1.4. If adiabatic elasticity of argon at pressure P is E , at what pressure the adiabatic elasticity of hydrogen will also equal to E?
A) $P$
B) $1.4P$
C) $\dfrac{7}{8}P$
D) $\dfrac{8}{7}P$

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Last updated date: 18th Jul 2024
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Answer
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Hint: In an adiabatic process, heat is neither added to the system nor it escapes the system. The pressure at which this happens is called the adiabatic pressure. To obtain adiabatic pressure, a number known as adiabatic constant $\gamma $ should be multiplied.

Complete step by step solution:
The adiabatic elasticity of a gas is given by-
$E = \gamma P$ Where E=elasticity, $\gamma $= ratio of two specific heats,
P= pressure of the gas.
For argon given that ${\gamma _{Ar}} = {\left( {\dfrac{{{C_P}}}{{{C_V}}}} \right)_{Ar}} = 1.6$
Therefore,
$E = {\gamma _{Ar}}P$
$ \Rightarrow E = 1.6P$
Let the pressure required for hydrogen is ${P'}$
Now given that
 ${\gamma _{{H_2}}} = {\left( {\dfrac{{{C_P}}}{{{C_V}}}} \right)_{_{{H_2}}}} = 1.4$
As elasticity of hydrogen is also E, So
$E = {\gamma _{{H_2}}}{P_0}$
$ \Rightarrow E = 1.4{P_0}$
Now compare the above equations, we get –
$1.4{P_0} = 1.6P$
$ \Rightarrow {P_0} = \dfrac{{1.6}}{{1.4}}P$
$ \Rightarrow {P_0} = \dfrac{8}{7}P$

Hence, the correct option is Option C.

Additional information: An adiabatic process is defined as, The thermodynamic process in which there is no exchange of heat from the system to its surrounding neither during expansion nor during compression.
The adiabatic process can be either reversible or irreversible.
Following are the essential conditions for the adiabatic process to take place:
1. The system must be perfectly insulated from the surrounding.
2. The process must be carried out quickly so that there is a sufficient amount of time for heat transfer to take place.

Note: The students should always remember that the value of ${C_P} > {C_V}$ and the relation between these two specific heats is given by Mayor’s Formula i.e. ${C_P} - {C_V} = R$ where R= universal gas constant.