If the specific heat capacity depend upon the temperature, what formula we should use instead of $Q=mc\Delta T$

Question

Recently, I learned that specific heat capacity is not constant in different temperatures and it depends on the temperature.If I have diagram like this

(consider that the red part is a quarter of ellipse)

1-How much energy(in calories) do we need to change the temperature of 1 kilogram of metal A from 0 to 40 Celsius?

2-How can we choose a single value (as an average value) for the specific heat capacity of metal?

Answer 1

1 You could find a relation between C and T from the assumptions(quarter of an ellipse) and just substitute that in the integral

$$ m\int C dT $$

2 We usually say specific heat capacity of metal is some value at particular temperature and pressure say ( Standard Temperature and Pressure ). When we know that the specific heat changes with temperature then we cant assume a single value ,if we know how C varies with T. But if the variation is negligible through the operating temperature then we can assume an average value. If you have say 10 points on a graph of C vs T, you could either average the values of C directly(if variation small) or find a relation between C and T from the points through regression analysis

Hope this helps

Answer 2

1-How much energy(in calories) do we need to change the temperature of 1 kilogram of metal A from 0 to 40 Celsius?

You simply integrate your function (find the area under the graph).

I guess, what you have on the y-axis is $c=\frac{Q}{m \Delta T}$. Integrating gives you the amount of energy per mass, $Q/m$.

2-How can we choose a single value (as an average value) for the specific heat capacity of metal?

Usually specific capacities are given for certain ranges of temperatures and pressures. Having one single value might give a very high uncertainty. Better would be to do the calculations in steps.

In your sketch an average value - e.g. by by setting $c=0.43$ - can give you a not perfectly precise value for the calculation, but depending on your required level uncertainty it might be fine.