Solution:

According to the statement, a mass m₁ of a substance A having specific heat capacity c₁ at temperature T₁ is mixed with a mass m₂ of other substance B having specific heat capacity c₂ at a lower temperature T₂. Due to this, the final temperature of the mixture becomes T.

Therefore, decrement in temperature of substance A = T₁ – T

Increment in the temperature of substance B = T – T₂

Using the expression of heat energy, we will now calculate the heat energy gained and lost by B and A respectively.

Heat energy lost by A = m₁ × c₁ × decrement in temperature

= m₁c₁ (T₁ – T)

Heat energy gained by B = m₂ × c₂ × increment in temperature

= m₂c₂ (T – T₂)

In the case where no energy goes out in the surrounding, then by the principle of mixtures we can write –

Heat energy lost by A will be equal to the Heat energy gained by B

m₁c₁ (T₁ – T) = m₂c₂ (T – T₂)

Upon rearranging the above equation, we have –

T = (m₁c₁ T₁ + m₂c₂ T₂) / m₁c₁ + m₂c₂

We have made an assumption here that no loss of heat energy takes place.