Question

A water molecule has its three atoms arranged in a "$V$" shape, so it has rotational kinetic energy around any of three mutually perpendicular axes. However, like diatomic molecules, its vibrational modes are not active at temperatures below $1000K$. What is the thermal energy of $2.0mol$ of steam at a temperature of $160°C$?

Short answer

The thermal energy is$50kJ$.

Step-by-step solution

Definition of thermal energy

The energy that a thing or area retains as a result of particle movement inside the object or system is known as thermal energy.

Thermal energy is one of several types of energy, with "energy" referring to the ability to do work. The movement of an element as a result of applied forces is referred to as work.

A system is more than just a collection of objects enclosed within a defined boundary. As a result, thermal energy is defined as an item's ability to perform work when particles are moving.

Step 2:Explanation

The water molecule has three translational degrees of freedom because its centre of mass can travel independently in any of the three directions. It has three rotational degrees of freedom because it includes spinning kinetic energy around any of three perpendicular axes.

. Under $1000K$it has no vibrational degrees of freedom. Therefore, the water molecule has in total $i=6$degrees of freedom. The number of degrees of freedom determines the molar specific heat at constant volume.

$C_V=\frac{i}{2}RT=3RT$

The value of thermodynamic energy 

The following formula is used to determine thermal energy.

$E_{\text{th}}=n{C}_{V}T=3nRT$

where $n$is the number of moles. Substituting numerical values we obtain

$E_{\text{th}}=50\mathrm{kJ}$