# Specific Heat Capacity Dependency on Temperature

Specific heat capacity depends on temperature.

When heating a substance, energy change = mass of substance $$\times$$ specific heat capacity $$\times$$ temperature change.

In school, we may apply this equation over a change in temperature to find the energy put into the system. At the same time, the value for the specific heat capacity is usually a constant.

Is this actually allowed? Surely if the SHC depends on temperature, then using a constant value for the SHC over a temperature change won't give a correct answer?

How does one choose the value for the SHC to use?

For example, if I want to know the energy required to change water from a temperature of $$10$$ degrees to $$50$$ degrees, then which value for the SHC of water would I use from this table (in the pictures)?

[Left column shows values for the SHC (right two columns) at the different temperatures in different units.] • You mean the LHC is the temperature, right? – Gert Aug 20 '20 at 16:47

## 1 Answer

For example, if I want to know the energy required to change water from a temperature of 10 degrees to 50 degrees, then which value for the SHC of water would I use from this table (in the pictures)?

Calculate the weighted average, $$SHC_{average}$$:

$$SHC_{average}=\frac{(20-10)\times 4.1910 +(25-20)\times 4.1570+(30-25)\times 4.1379 +(40-30)\times 4.1175 +(50-40)\times 4.0737}{10+5+5+10+10}$$ $$SHC_{average}=4.137$$

Or more generally:

$$SHC_{average}=\frac{\displaystyle\sum_{i=1}^n\Big(SHC_i \times \Delta T_i\Big)}{\displaystyle\sum_{i=1}^n \Delta T_i}$$

where $$SHC_i$$ is the SHC of interval $$i$$, and $$\Delta T_i$$ the corresponding temperature interval. $$n$$ is the number of intervals.

Alternatively (but more laboriously) you could curve-fit a function $$f(T)$$ to the data, by means of regression analysis. Depending on the data-points a linear ($$a_0+a_1 T$$) or quadratic ($$a_0+a_1 T+a_2 T^2$$) model should probably suffice.

The heat required to go from $$T_1$$ to $$T_2$$ is then calculated by integration:

$$Q_{1 \to 2}=m\int_{T_1}^{T_2}f(T)\text{d}T$$