# What is the effect of heat on the inertial mass of a body?

1. Nov 21, 2015

### tejas sanap

I mean, a body with higher temperature has higher energy so is it heavier than before? We know from e=m*c^2 that energy is equivalent to mass

2. Nov 21, 2015

### Drakkith

Staff Emeritus
It is indeed heavier when heated than it was when it was cooler! The difference is just too little to notice in our everyday lives.

3. Nov 21, 2015

### Geofleur

Consider a balloon full of a gas. We can write the energy of the balloon as a sum of the energies of all the gas molecules inside it (let's ignore the potential and kinetic energy in the rubber of the balloon itself). We can use Einstein's formula to write the energy of each molecule of mass $m_i$ as

$E_i = \frac{m_i c^2}{\sqrt{1-\frac{v_i^2}{c^2}}}$,

where $v_i$ is the speed of the i$^{th}$ molecule. The extra factor on the bottom takes into account that the gas molecules are zipping around inside the balloon. The form of the equation that everybody knows, $E = mc^2$ is really written for an object that is not moving in the frame of reference where $E$ is being measured (well, some people used to represent $\frac{m_i}{ \sqrt{1-v_i^2/c^2}}$ as a "new" mass, say M_i, that changes with a molecule's velocity, and write $E_i = M_i c^2$, but that way of doing things does not seem as popular as it used to be).

The molecules in the balloon are really going fast compared to objects that we are used to in everyday life, but they are not going fast at all compared to the speed of light. We can use calculus to show that, in this case, the energy of each molecule can be approximately re-written as

$E_i = \frac{m_i c^2}{\sqrt{1-\frac{v_i^2}{c^2}}} \approx m_i c^2 + \frac{1}{2}m_i v_i^2$.

You may notice that the second term is the molecule's kinetic energy. The total energy of the gas in the balloon can be gotten by adding up all these energies. When the gas in the balloon is heated, it makes the molecules zip around faster. That makes the energy above bigger for each molecule (because $v_i$ gets bigger), which makes the energy of the whole balloon bigger. But for the balloon as a whole, we can write $E = M_{gas} c^2$. We have

$E = M_{gas} c^2 = E_1 + E_2 + \cdots E_N$,

assuming there are $N$ molecules in the balloon. Now here is something really neat. If we divide both sides of the above equation by $c^2$, and write everything out explicitly, we get

$M_{gas} = m_1 + m_2 + \cdots m_N + \frac{1}{2}m_1\frac{v_1^2}{c^2}+ \cdots \frac{1}{2}m_N\frac{v_N^2}{c^2}$.

That is to say, the mass of the gas in the balloon is not the sum of the masses of each of the molecules - the kinetic energy terms have been added on! This formula let's us see explicitly why the balloon's mass increases when we heat it up.