# The temperature, T, of a given mass of gas, varies inversely with its volume, V. The temperature...

## Question:

The temperature, T, of a given mass of gas, varies inversely with its volume, V. The temperature of 20 {eq}cm^3 {/eq} of a certain gas is {eq}15^\circ C {/eq}. What will the temperature be when it is compressed to a volume of 5 {eq}cm^3 {/eq}?

## Inverse Variation:

The inverse variation shows the relationship between two variables that change or vary in the opposite direction. For such variables, their product is equal to a constant. This constant is referred to as the constant of variation.

If the temperature of a gas varies inversely with its volume, we can write this as:

• {eq}T \propto \dfrac{1}{V} {/eq}

To remove the proportionality sign, we replace it with with an equal sign and add a constant of variation.

• {eq}T = \dfrac{k}{V} {/eq}

Given that the temperature is {eq}15^o {/eq} when the volume is {eq}20\rm cm^3 {/eq}, then:

• {eq}15 = \dfrac{k}{20} {/eq}

Solving for k:

• {eq}k = 15\times 20 = 300 {/eq}

Therefore, the equation representing the relationship between temperature and volume is:

• {eq}T = \dfrac{300}{V} {/eq}

Using the above equation, the temperature of the gas when it is compressed to {eq}V = 5\rm cm^3 {/eq} is equal to:

• {eq}T = \dfrac{300}{5} = \boxed{60^o} {/eq} 