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Problem 3.102:A cylinder of fixed capacity $44.8$ liters contains helium gas at standard temperature and pressure. What is the amount of heat needed to raise the temperature of the gas in the cylinder by $15\:^oC$. Take $R$ as $8.314\:Jmol^{-1}$

Solution:

The gas is in a cylinder of fixed capacity. So the process is isochoric in nature.

The cylinder has a capacity of $44.8$ liters.

The gas is at standard temperature and pressure.

From Avogadro's law, $22.4$ liters of gas at standard temperature and pressure will be one mole.

So the given gas has $2$ mole.

Helium is monoatomic with degree of freedom $f\:=\:3$.

Specific heat capacity at constant volume, $C_v\:=\:\frac{f}{2}R$.

With $f\:=\:3$, $C_v\:=\:\frac{3}{2}R$.

The required change in temperature of the gas in the cylinder is $15\:^oC$

This is equal to change of $15\:K$.

Heat required for this constant volume process can be estimated using the relation $Q\:=\:nC_v\delta T$.

Then, $Q\:=\:2\times\frac{3}{2}R\times 15$.

Or, $Q\:=\:3\times R\times 15\:=\:3\times 8.314\times 15\:=\:374.13\:J$.

Therefore, the amount of heat needed to raise the temperature of  $44.8$ liters helium gas without changing its volume is $374.13\:J$.