# Mass from specific heat?

1. Aug 25, 2004

### Zorodius

A problem in my book asks me to compute the mass of a gas molecule given its specific heat at constant volume. ("Take $c_v$ = 0.075 cal/g C° for argon and calculate the mass of an argon atom and the molar mass of argon.")

Can I get a hint on where to go with this? I don't see any way of determining the mass when I only know the specific heat under constant volume, although it makes sense that there would be one.

2. Aug 25, 2004

### ehild

Internal energy of ideal gases - the argon gas is monoatomic - Equipartition Principle - Boltzmann's Constant - Avogadro's Number ...

ehild

3. Aug 27, 2004

### ehild

some more hints.

Imagine a vessel with a piston. The vessel is filled with argon gas. You can give energy to the gas either by heating or exerting work on it.
$$\Delta U = \delta Q + \delta W$$.
The work means (we assume only mechanical work) that we exert force on the piston and move it. But we are interested in a process with constant volume. So the energy changes by adding some amount of heat. The specific heat is the amount of heat energy wich rises the temperature of unit mass of a substance by 1 degree (celsius or kelvin). At constant volume
$$\Delta U = c_v*m*\Delta T$$ or
$$c_v = \frac{1}{m}*\frac{\partial U}{ \partial t}|_V$$.
An ideal gas consist of atoms or molecules interacting only with the walls of the container when they collide to it. The internal energy of an ensemble of non-interacting particles is the sum of their KE. The Equipartition Principle states that the average KE per degrees of freedom is
$$\frac{1}{2} \kappa T$$
($$\kappa$$ is Boltzmann's Constant). Argon is a noble gas, it is monoatomic. The degrees of freedom of a single atom is 3. So the average energy of an argon atom is
$$\frac {3}{2} \kappa T$$
The energy of an ensemble of gas, containing N atoms is
$$U = \frac{3}{2}N \kappa T$$.
If the mass of one argon atom is m, 1 g argon contains N=1/m atoms. The specific heat of the argon gas is therefore

$$c_v= \frac{3}{2}\kappa \frac{1}{m}$$.

Knowing cv, you can calculate the mass of one atom. Take care of the units.
The atomic mass is the mass of one atom multiplied by Avogadro's number, $$N_A$$.

ehild

4. Aug 27, 2004

### Locrian

Good stuff ehild, thanks.