Question

Substance A has a large specific heat (on a per gram basis), while substance B has a smaller specific heat. If the same amount of energy is put into a 100 g block of each substance, and if both blocks were initially at the same temperature, which one will now have the higher temperature?

Short answer

The temperature of Substance B will be higher.

Step-by-step solution

Definition of specific heat

The specific heat capacity is the amount of heat energy which is required to raise the temperature of a substance per unit of mass.

Finding thermal energy equation

To begin, create the thermal energy equation, in which we solve for substance A's specific heat (CA).

Write the equation for thermal energy and solve for substance A.

$\begin{array}{c}\mathrm{E}={\mathrm{mC}}_{\mathrm{A}}{\mathrm{\Delta T}}_{\mathrm{A}}\\ ={\mathrm{mC}}_{\mathrm{A}}({\mathrm{T}}_{\mathrm{f},\mathrm{A}}-{\mathrm{T}}_{\mathrm{i}})\\ {\mathrm{C}}_{\mathrm{A}}=\frac{\mathrm{E}}{\mathrm{m}({\mathrm{T}}_{\mathrm{f},\mathrm{A}}-{\mathrm{T}}_{\mathrm{i}})}\end{array}$

Similarly, ${\mathrm{C}}_{\mathrm{B}}=\frac{\mathrm{E}}{\mathrm{m}({\mathrm{T}}_{\mathrm{f},\mathrm{B}}-{\mathrm{T}}_{\mathrm{i}})}$

Finding the temperature of the substance

To show that the fact that CA is greater than CB implies that substance B has a higher ultimate temperature than substance A.

$\begin{array}{c}{\mathrm{C}}_{\mathrm{A}}>{\mathrm{C}}_{\mathrm{B}}\\ \frac{\mathrm{E}}{\mathrm{m}({\mathrm{T}}_{\mathrm{f},\mathrm{A}}-{\mathrm{T}}_{\mathrm{i}})}>\frac{\mathrm{E}}{\mathrm{m}({\mathrm{T}}_{\mathrm{f},\mathrm{B}}-{\mathrm{T}}_{\mathrm{i}})}\\ {\mathrm{T}}_{\mathrm{f},\mathrm{B}}-{\mathrm{T}}_{\mathrm{i}}>{\mathrm{T}}_{\mathrm{f},\mathrm{A}}-{\mathrm{T}}_{\mathrm{i}}\\ {\mathrm{T}}_{\mathrm{f},\mathrm{B}}>{\mathrm{T}}_{\mathrm{f},\mathrm{A}}\end{array}$

Therefore, the temperature of Substance B will be higher.