Q.When a cycle tyre suddenly bursts, the air inside the tyre expands. This process is

An adiabatic process is one in which no heat flows into or out of the system at all ($Q=0$), so the first law reduces to $\Delta U = -W$: the gas can only expand at the expense of its own internal energy (cooling as it does so), or be compressed by external work done on it (heating as a result). This can happen either because the system is thermally insulated from its surroundings, or because the process happens so fast that there simply isn't time for heat to flow even without insulation (e.g. a bursting tyre, or the rapid compression stroke of a diesel engine, or warm air rising and adiabatically expanding and cooling to form clouds). The equation of state for a quasi-static adiabatic process is $PV^\gamma = \text{constant}$, where $\gamma=C_p/C_v$ is the adiabatic exponent; equivalently $TV^{\gamma-1}=\text{constant}$ or $T^\gamma P^{1-\gamma}=\text{constant}$. On a P-V diagram the adiabatic curve (adiabat) is always steeper than the isotherm through the same point, because $\gamma>1$. The work done during an adiabatic process from $(T_i,V_i)$ to $(T_f,V_f)$ works out to $W_{adia} = \dfrac{\mu R}{\gamma-1}(T_i-T_f),$ positive (gas cools) during adiabatic expansion and negative (gas heats) during adiabatic compression.