Question

A diatomic ideal gas initially at 273K is given 100 calorie heat due to which system did 209 Joule work. Molar heat capacity ( ${C_m}$ ) of gas for the process is:
A.\[\dfrac{3}{2}R\]
B.\[\dfrac{5}{2}R\]
C.\[\dfrac{5}{4}R\]
D.\[5R\]

Answer 1

Hint:We have been given work done and heat energy given to the system. Using first law of thermodynamics, we can get Change in internal energy, by using formula:
\[\Delta U = q + W\]
After we get change in internal energy, we know the formula for change in internal energy is given by:
\[\Delta U = {C_v}\Delta T\]
Now, from the above equation we get, change in temperature. Then, using change in heat energy formula we can easily get ${C_m}$ using the below formula:
$\Delta Q = {C_m}\Delta T$

Complete step by step answer:
Molar Specific heat capacity (${C_m}$) is defined as heat required to raise the temperature of 1 mole substance by 1 kelvin temperature.
Heat given to the system will be positive. It is denoted by q:
\[q = 100cal\] (given in question)
But to use, we need all values in Joules, so we have to convert calorie into joules.
\[1cal = 4.184J\]
\[\therefore q = 100 \times 4.184 = 418.4J\]
Since, Work is done by the system on surroundings, so it will be negative. Work done is given by W:
\[\therefore W = - 209J\]
Now using the first law of thermodynamics, which is the law of conservation of energy states that energy can neither be created nor be destroyed, it can only change from one form to another.
So, change in internal energy will be the result of heat given to the system and work done on the system. Mathematically we can write as:
\[\Delta U = q + W\]
Here, \[\Delta U = {\text{ Change in Internal Energy}}\], q is heat given to the system and W is work done.
Substitute the values to get change in internal energy,
\[\Delta U = 418.4 - 209\]
Calculating and simplifying we get,
\[\therefore \Delta U = 209.4J\]
We know for diatomic molecule, \[{C_v} = \dfrac{5}{2}R\]
Here \[{C_v}\] is molar heat capacity at constant volume and R is Universal Gas constant.
\[\Delta U = {C_v}\Delta T\]
\[\Delta T\] is change in temperature, Substitute the values to get change in temperature:
\[209.4 = \dfrac{5}{2}R \times \Delta T\]
Taking all other values on one side, we can calculate change in temperature,
\[\Delta T = \dfrac{{209.4 \times 2}}{{5R}}\]
Now, we know heat exchange formula can be written as:
\[\Delta Q = {C_m}\Delta T\]
Here, \[\Delta Q\] is heat change, \[{C_m}\] is molar Heat Capacity of gas.
We know from given data,
\[\Delta Q = 418.4J\]
Substitute all values to get molar heat capacity as:
\[418.4 = {C_m} \times \dfrac{{209.4 \times 2}}{{5R}}\]
Taking all numerical values on one side, we get:
\[{C_m} = \dfrac{{5R \times 418.4}}{{209.4 \times 2}}\]
On simplification, we can calculate the value to be approximate to:
\[\therefore {C_m} \approx 5R\]
Thus, the correct option is (D) i.e 5R.

Note:
Here R is universal gas constant and we need not substitute the value of it as the final options contain R in it.
Do not use the values of work done and heat exchange in different units, it should be in the same units and both can be converted to joules.
Don’t be confused with molar heat capacity and specific heat capacity, as molar heat capacity is heat required to raise the temperature of 1 mole substance by 1 kelvin temperature, while Specific heat capacity is heat required to raise the temperature of 1 Kg substance by 1 kelvin temperature. Here we have to use and find molar heat capacity (${C_m}$ ).