A diatomic ideal gas is compressed adiabatically to of its initial volume. If the initial temperature of the gas is (in Kelvin) and the final temperature is , the value of is ______. [2010]

Question

A diatomic ideal gas is compressed adiabatically to $\frac{1}{32}$ of its initial volume. If the initial temperature of the gas is $T_{i}$ (in Kelvin) and the final temperature is $a T_{i}$ , the value of $a$ is ______. [2010]

Smart Achievers · Accepted Answer

(4)

For an adiabatic process, applying $T V^{γ - 1} = constant$

$a$

$γ = 1.4 (for diatomic gas), V_{2} = \frac{V_{1}}{32}, T_{1} = T_{i}, T_{2} = a T_{i}$

$∴$ $T_{i} = a T_{i} {[\frac{1}{32}]}^{1.4 - 1} = a T_{i} {[\frac{1}{2^{5}}]}^{0.4} = \frac{a T_{i}}{4}$

$∴$ $a = 4$

Solution

Q.
A diatomic ideal gas is compressed adiabatically to $\frac{1}{32}$ of its initial volume. If the initial temperature of the gas is $T_{i}$ (in Kelvin) and the final temperature is $a T_{i}$ , the value of $a$ is ______. [2010]

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Solution

Q. A diatomic ideal gas is compressed adiabatically to 132 of its initial volume. If the initial temperature of the gas is Ti (in Kelvin) and the final temperature is aTi, the value of a is ______. [2010]

Ans. (4) For an adiabatic process, applying TVγ-1=constant T1V1γ-1=T2V2γ-1⇒T1=T2(V2V1)γ-1 γ=1.4 (for diatomic gas), V2=V132, T1=Ti, T2=aTi ∴ Ti=aTi[132]1.4-1=aTi[125]0.4=aTi4 ∴ a=4

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Q.
A diatomic ideal gas is compressed adiabatically to $\frac{1}{32}$ of its initial volume. If the initial temperature of the gas is $T_{i}$ (in Kelvin) and the final temperature is $a T_{i}$ , the value of $a$ is ______. [2010]