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.