# Relation Between Density and Temperature

Temperature is the measure of heat. Density is the measure of how closely any given entity is packed or it is the ratio of the mass of the entity to its volume. The relation between density and temperature is inversely proportional. Change in density will be reflected in a change in temperature and vise-versa.

## Density and Temperature

The density and temperature relationship for ideal gases is mathematically written as-

 Formula Terms SI units For ideal gas $P=\rho RT$ P is the pressure of the ideal gas pascal or Pa R is the universal gas constant R=8.31 J/mole/K0 T is the temperature of the ideal gas Kelvin or K0 $\rho$ is the density of the ideal gas. Kg/m3

## Density and Temperature Relationship

The density and temperature relation are proportionate. That is, the density is inversely proportional to temperature. Which means for unit volume-

• When density increases, the temperature decrease.
• When density decreases, temperature increases.
• When more temperature increases, density reduces.
• Whenthe temperature decrease, density increases.

## Density and Temperature Equation

Deriving Density and Temperature Equation is very important to understand the concept. Below is the derivation of Density and Temperature relation for the ideal gas.

### Equation of state ideal gas

In thermodynamics, the relation between Density and Temperature is expressed through Equation of states for ideal gases. Consider an ideal gas with-

• Pressure P
• Volume V
• Density $\rho$
• Temperature T
• Universal gas constant R
• Number of moles n

Applying Boyle’s law and Charles and Gay-Lussac law we get-

• Boyle’s law: For a given mass, at a constant temperature, the pressure times volume is constant.

PV = C1
• Charles and Gay-Lussac law: For a given mass, at constant pressure, the volume is directly proportional to the temperature.

V = C2T

Combining both we get-

$\frac{PV}{T}=nR$ $\Rightarrow PV=nRT$

Dividing both sides by mass m we get-

$\Rightarrow \frac{PV}{m}=\frac{nRT}{m}$———–(1)

Here, Specific volume(v) can be defined as the ratio of volume to its mass. That is $v=\frac{Volume}{Mass}=\frac{V}{m}=\frac{1}{\rho }$

Substituting specific volume in equation(1) we get-

$\Rightarrow Pv=\frac{nRT}{M}$

Or

Pv = RT

Or

$P=\frac{RT}{v}$ $\Rightarrow P=\rho RT$

Hope you understood the relation between Density and Temperature in Thermodynamics.

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