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## Homework Statement

At low temperatures, the specific heats of metals is described by the expression

## c=kT + AT^3 ##

, where k and A are constants. Here the first term describes the contribution of free electrons and the second the lattice contribution.

How much heat is required to raise the temperature of 1 g of potassium from 1 K to 5 K?

For potassium, ## k=\frac{2.1mJ}{(mol K^2)} \\ \\ \\ A=\frac{2.6mJ}{(mol K^4)} ##

## Homework Equations

Above

##M##gram ##=\frac{1}{\mu}##mol

## The Attempt at a Solution

## C= \frac{\Delta Q}{\Delta T M} ##

Wheareas: ##\Delta Q## is the change in the amount of heat , ##\Delta T## is the change in the temperature.

## \Delta Q = CM \Delta T = (kT+AT^3)M\Delta T ##

## \int dQ = \frac{1}{\mu} \int ^{T_2} _{T_1} (kT+AT^3)dT ##

## Q = \frac{1}{\mu} (\frac{KT^2}{2} + \frac{AT^4}{4})^{T_2} _{T_1} ##

##T_1 = 1## ##T_2 = 5## (is this correct?)

Did I use the formula correctly , if so , could you help me with the the integration factors please.

Thanks in advance.