In the complete combustion of butanol $C_4H_9OH_{(l)}$,if $\Delta H$ is the enthalpy of combustion and $\Delta E$ is the heat of combustion at constant volume,then

$A$ gas expands from a volume of $1 \text{ dm}^3$ to $1.25 \text{ dm}^3$ against a constant external pressure of $1 \text{ bar}$. Calculate the change in internal energy $(\Delta U)$ if $150 \text{ J}$ of heat is supplied to the system. (in $\text{ J}$)

For water at $100^{\circ} C$ and $1 \, bar$,$\Delta_{vap} H - \Delta_{vap} U = ...... \times 10^{2} \, J \, mol^{-1}$. (Round off to the Nearest Integer) $[Use : R = 8.31 \, J \, mol^{-1} \, K^{-1}]$ [Assume volume of $H_{2}O_{(\ell)}$ is much smaller than volume of $H_{2}O_{(g)}$. Assume $H_{2}O_{(g)}$ treated as an ideal gas]

One mole of gas absorbs $200 \ J$ of heat at constant volume,then its temperature rises from $298 \ K$ to $308 \ K$. The change in internal energy will be $... \ J$.

$A$ system performs mechanical work equal to $15 \ kJ$ and loses $2 \ kJ$ of heat to the surroundings. What is the change in internal energy of the system (in $kJ$)?

Give the mathematical expression for the first law of thermodynamics.