Question

Equation of gas in terms of pressure (p), absolute temperature (T) and density (d) is
A) $\dfrac{{{p_1}}}{{{d_1}{T_1}}} = \dfrac{{{p_2}}}{{{d_2}{T_2}}}$
B) $\dfrac{{{p_1}{T_1}}}{{{d_1}}} = \dfrac{{{p_2}{T_2}}}{{{d_2}}}$
C) $\dfrac{{{p_1}{d_2}}}{{{T_1}}} = \dfrac{{{p_2}{d_1}}}{{{T_2}}}$
D) $\dfrac{{{p_1}{d_1}}}{{{T_1}}} = \dfrac{{{p_2}{d_2}}}{{{T_2}}}$

Answer 1

We can use the equation for ideal gas given by the ideal gas law and then using the given quantities of temperature, pressure and density which relationship amongst them is constant throughout. The constant quantity will be the same before and after the changes undergone and thus will give us the required option.

Formula used:
$PV = nRT$ where, P is the pressure, V is the volume, R is the universal gas constant, n is the number of moles and T is the temperature.

Complete step by step answer:
The given pressure, temperature and density are p, T and d respectively. Let its mass be m and molecular mass be M.
According to the ideal gas law, the equation for ideal gas is given as:
$pV = nRT \\\ \Rightarrow p = \dfrac{n}{V}RT.....(1) \\\$
Where,
p is the pressure, n is the number of moles of gas, R is universal gas constant and T is the temperature.
Number of moles (n) is equal to the mass of the gas (m) divided by its molecular mass (M)
$\Rightarrow n = \dfrac{m}{M}$
Substituting this value in (1), we get:
$p = \dfrac{m}{{M \times V}}RT$
The density of the gas is its mass per unit volume and for the given gas:
$d = \dfrac{m}{V}$
So the equation becomes:
$p = \dfrac{d}{M}RT \\\ \Rightarrow \dfrac{p}{{dT}} = RM \\\$
On the R.H.S of this equation, R is the universal constant and M is the molecular mass of the gas which will not be changed and thus is considered at a constant quantity.
$\dfrac{p}{{dT}} = const.$
It means that this quantity after the changes undergone by gas remains the same.
$\dfrac{{{p_1}}}{{{d_1}{T_1}}} = \dfrac{{{p_2}}}{{{d_2}{T_2}}}$
Depicting that this factor before and after the changes remains the same.

So, the correct answer is “Option D”.

Note:
We have to see what the quantities are given and what replacements need to be done so that we can get desired quantities in the equation.
We found a constant relationship because from the question we can say that both the states of these physical quantities are equal and this will only be possible when their relationship would be constant.