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# An ideal gas on heating from $100{{K}}$ to $109{{K}}$ shows an increase by ${{a}}\%$ in its volume at constant ${{P}}$. The value of ${{a}}$ is …………

Last updated date: 17th Sep 2024
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Hint: Ideal gas is a type of imaginary gas in which the particles are randomly moving and they do not interact with each other. They are point-like particles. Also, they obey the gas law which is the combination of Boyle’s law, Charles’ law, Gay Lussac law and Avogadro law.

It is given that the initial temperature, ${{{T}}_1} = 100{{K}}$
Final temperature, ${{{T}}_2} = 109{{K}}$
Percentage increase in volume, ${{V\% = a\% }}$
According to ideal gas equation, ${{PV}} = {{nRT}}$, where ${{P}}$ is the pressure of the gas, ${{V}}$ is the volume of gas, ${{n}}$ is the number of moles of gas, ${{R}}$ is the gas constant and ${{T}}$ is the temperature.
Here, ${{P}},{{n}},{{R}}$ are constants. Since the same ideal gas is heated, there is no change in the number of moles of the gas. Pressure is given as constant.
Thus we can write the ideal gas equation as:
$\dfrac{{{{{V}}_2}}}{{{{{V}}_1}}} = \dfrac{{{{{T}}_2}}}{{{{{T}}_1}}}$, where ${{{V}}_1}$ and ${{{V}}_2}$ are the initial and final volumes of gas respectively.
Substituting the values of initial and final temperatures, we get
$\dfrac{{{{{V}}_2}}}{{{{{V}}_1}}} = \dfrac{{109}}{{100}} \Leftrightarrow {{{V}}_2} = 1.09 \times {{{V}}_1}$
Here, the percentage increase in the volume can be expressed as
$\dfrac{{{{{V}}_2} - {{{V}}_1}}}{{{{{V}}_1}}} \times 100 = {{a}}$
Substituting the value of ${{{V}}_2}$ in the above equation, we get
$\dfrac{{1.09{{{V}}_1} - {{{V}}_1}}}{{{{{V}}_1}}} \times 100 = \dfrac{{9{{{V}}_1}}}{{{{{V}}_1}}} = 9$
Thus the percentage increase in the volume, ${{a}}\% = 9\%$.

So, the value of ${{a}}$ is $9$

Note: Ideal gas equation involves four equations which explain the relationship between volume and pressure at constant number of moles and temperature (Boyle’s law); volume and temperature at constant pressure and number of moles (Charles’ law); pressure and temperature at constant number of moles and volume (Gay Lussac law); number of moles and volume at constant pressure and temperature (Avogadro law).