Question

One mole of an ideal gas requires 207 J heat to raise the temperature by 10K when heated at constant pressure. If the same gas is heated at constant volume to raise the temperature by the same 10K, the heat required is (R=8.3 J/mol-K)
A) 187 J
B) 29 J
C) 215.3 J
D) 124 J

Answer 1

Hint
Here, we have to find the value of heat required. For this firstly we use the formula of heat at constant pressure i.e. $Q = n{C_P}\Delta T$. By substituting the given values we can find the value of Cp. after that we have to calculate the value of Q at constant volume for this we need to calculate the value of CV for this we use the formula ${C_P} = {C_V} + R$, on substituting the values we will get the heat at constant volume.

Complete Step by step solution
Here, it is given that one mole of an ideal gas requires $207 J$ heat to raise the temperature by 10K when heated at constant pressure. So, using the formula of heat at constant pressure is
$Q = n{C_P}\Delta T$
Where, $n$ is the number of moles,
$C_P$ is the specific heat at constant pressure,
$∆T$ is the change in temperature.
As, $n = 1 mole, Q = 207 J, ∆T = 10K$
On substituting the values, we get
$ \Rightarrow 207 = {C_P} \times 10$
$ \Rightarrow {C_P} = 20.7$
Now, as we know that the relation between the specific heat at constant volume and at constant pressure is ${C_P} = {C_V} + R$, $R$ is the gas constant and it is given as $R=8.3 J/mol-K$.
On substituting the value of $C_P$, we get
$ \Rightarrow {C_V} = 20.7 - 8.3 = 12.4$
Now, the heat required to raise the temperature of gas by $10K$ at constant volume is
$Q = n{C_V}\Delta T$
On substituting the values, we get
$Q = 12.4 \times 10 = 124 J$
Hence, (D) option is correct.

Note
Specific heat is defined by the amount of heat needed to raise the temperature of 1 gram of a substance 1 degree Celsius (°C). Water has a high specific heat capacity which we'll refer to as simply "heat capacity", meaning it takes more energy to increase the temperature of water compared to other substances. Its formula is $Q = mC\Delta T$.