Prove that: i. Cv = [dE/dT]v ii. Cp = [dH/dT]p

The change in internal energy ($E$) for a system is given by:

$$dE = \delta q + \delta w$$

where $\delta q$ is the heat absorbed by the system and $\delta w$ is the work done on the system. For a reversible process, the work done is $\delta w = -P dV$.
Substituting this into the First Law:

$$dE = \delta q - P dV$$

If the volume ($V$) is kept constant, then $dV = 0$. In this case, the work done is zero ($\delta w = 0$).
The First Law simplifies to:

$$dE = \delta q_V$$

where $\delta q_V$ is the heat absorbed at constant volume.

Heat capacity at constant volume is defined as the amount of heat required to raise the temperature of a substance by one degree Celsius (or Kelvin) at constant volume:

$$C_V = \left(\frac{\delta q_V}{\partial T}\right)_V$$

Since $dE = \delta q_V$ at constant volume, we can replace $\delta q_V$ with $dE$:

$$C_V = \left(\frac{\partial E}{\partial T}\right)_V$$

This proves the first relation.

$$\boxed{C_V = \left[\frac{\partial E}{\partial T}\right]_V}$$

Enthalpy is defined as:

$$H = E + PV$$

where $E$ is internal energy, $P$ is pressure, and $V$ is volume.
Taking the total differential of $H$:

$$dH = dE + d(PV) = dE + P dV + V dP$$

From Step 1 of part (i), we know $dE = \delta q - P dV$.
Substitute this into the expression for $dH$:

$$dH = (\delta q - P dV) + P dV + V dP$$ $$dH = \delta q + V dP$$

If the pressure ($P$) is kept constant, then $dP = 0$.
The expression for $dH$ simplifies to:

$$dH = \delta q_P$$

where $\delta q_P$ is the heat absorbed at constant pressure.

Heat capacity at constant pressure is defined as the amount of heat required to raise the temperature of a substance by one degree Celsius (or Kelvin) at constant pressure:

$$C_P = \left(\frac{\delta q_P}{\partial T}\right)_P$$

Since $dH = \delta q_P$ at constant pressure, we can replace $\delta q_P$ with $dH$:

$$C_P = \left(\frac{\partial H}{\partial T}\right)_P$$

This proves the second relation.

$$\boxed{C_P = \left[\frac{\partial H}{\partial T}\right]_P}$$