# How much heat required to raise the temperature of potassium

1. Jun 7, 2016

### Asafish

1. The problem statement, all variables and given/known data
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, $k=\frac{2.1mJ}{(mol K^2)} \\ \\ \\ A=\frac{2.6mJ}{(mol K^4)}$

2. Relevant equations
Above
$M$gram $=\frac{1}{\mu}$mol
3. 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.