For one mole of an ideal gas, the amount of heat required to raise...

The heat required to raise the temperature of $n$ moles of an ideal gas by $\Delta T$ at constant volume is given by $Q_V = n C_V \Delta T$.

Given $n = 1\text{ mol}$ and $\Delta T = 1\text{ K}$, we have $Q_V = C_V = x$.

The heat required to raise the temperature of the gas by $\Delta T$ at constant pressure is given by $Q_P = n C_P \Delta T$.

For $n = 1\text{ mol}$ and $\Delta T = 1\text{ K}$, we have $Q_P = C_P$.

Using Mayer's relation for an ideal gas, $C_P - C_V = R$.

Substituting $C_V = x$, we get $C_P = x + R$.

Therefore, the amount of heat required at constant pressure is $x + R$.

Answer: $x + R$