Question 795: Thermodynamics & KTG

Question

The specific heat of argon at constant pressure and constant volume are $C_p$ and $C_v$ respectively. It's density ' $\rho$ ' at N.T.P. will be $[\mathrm{P}$ and T are pressure and temperature respectively at N.T.P.]

Accepted Answer

The density of a gas can be related to its pressure, temperature, and specific heat capacities through the ideal gas laws and the relation between specific heat capacities.
For an ideal gas, the relation between the specific heat capacities is given by:
$$ C_p - C_v = R $$
where $R$ is the universal gas constant. For argon at Normal Temperature and Pressure (N.T.P.), using the ideal gas equation:
$$ PV = nRT $$
where:
$ P $ is the pressure,
$ V $ is the volume,
$ n $ is the number of moles,
$ R $ is the gas constant,
$ T $ is the temperature.
The molar mass $M$ and the number of moles $ n $ are related to the density $ \rho $ as follows:
$$ \rho = \frac{m}{V} = \frac{nM}{V} = \frac{PM}{RT} $$
Since we're interested in the density $ \rho $ at N.T.P., we use the relation:
$$ \rho = \frac{P}{T \frac{R}{M}} = \frac{P}{T \frac{C_p - C_v}{M}} $$
which simplifies to:
$$ \rho = \frac{P}{T (C_p - C_v)} $$
Therefore, the correct option for the density of argon at N.T.P. in terms of the specific heat capacities is:
Option A:
$$ \frac{P}{T(C_p-C_v)} $$

Answer

$\frac{P}{T\left(C_p-C_v\right)}$

Answer

$\frac{\mathrm{PT}}{\left(\mathrm{C}_{\mathrm{p}}-\mathrm{C}_{\mathrm{v}}\right)}$

Answer

$\frac{T\left(C_p-C_v\right)}{P}$

Answer

$\frac{\left(\mathrm{C}_{\mathrm{p}}-\mathrm{C}_{\mathrm{v}}\right)}{\mathrm{PT}}$

Thermodynamics & KTG

The specific heat of argon at constant pressure and constant volume are C_p and C_v respectively. It's density ' \rho ' at N.T.P. will be [\mathrm{P} and T are pressure and temperature respectively at N.T.P.]

Choose the correct answer: