Question

The heat capacity per mole of water is (R is universal gas constant)
A. 9R
B. $${\displaystyle \frac{9}{2}}R$$
C. 6R
D. 5R

Answer 1

Hint: to solve this question we must know the definition of molar heat capacity and degree of freedom of gases and the value of degree of freedom of different type of gases should also be known us-
Molar heat capacity: it is defined as the amount of heat required to raise the temperature of one mole of gas by $$1^{\circ }C$$ (or 1K) and it is given as $$C={\displaystyle \frac{dQ}{dT}}$$ or $$C={\displaystyle \frac{dU}{dT}}$$ for one mole of gas.
Degree of freedom (f): it is the number of possible ways in which a gas molecule possesses kinetic energy. For nonlinear polyatomic gas value of degree of freedom is 6

Complete step by step solution
To find molar heat capacity we need amount of energy stored in mole of water at any temperature T as $$C={\displaystyle \frac{dU}{dT}}$$ the amount of energy stored by each atom in one mole of water is given as $$U={\displaystyle \frac{fRT}{2}}$$ and the total energy stored is $$U={\displaystyle \frac{3fRT}{2}}$$ as the total number of atom in water molecule is 3
For water as it is non-linear polyatomic f=6 so, $$U={\displaystyle \frac{18}{2}}RT=9RT$$
Now, $$C={\displaystyle \frac{dU}{dT}}$$ =$${\displaystyle \frac{d(9RT)}{dT}}=9R$$ and hence the option A will be correct.


Note:$$dQ$$ is the small amount of heat and $$dU$$ is the small change in internal energy and $$U={\displaystyle \frac{fRT}{2}}$$ is the average energy of one atom of the molecule and one more thing you have to keep in your mind that $$C={\displaystyle \frac{dQ}{dT}}$$ is the molar heat capacity of one mole for n mole it will be $$C={\displaystyle \frac{dQ}{ndT}}$$