Final answer:
By integrating the isobaric expansion coefficient and the isochoric pressure-temperature coefficient for the gas, and considering the conditions when T = T0, V = V0, and p = p0, we derive the ideal gas law (pV = RT) as the equation of state for the gas.
Step-by-step explanation:
We are tasked with finding the equation of state for a gas given the isobaric expansion coefficient (dV/dT = R/p) and the isochoric pressure-temperature coefficient (dp/dT = p/T). To find the equation of state, we integrate each of these coefficients. The isobaric expansion coefficient implies:
dV = (R/p) dT,
which upon integrating at constant pressure (p) gives us:
V = (R/p) T + V0,
where V0 is the integration constant, which can be interpreted as the volume when the temperature is absolute zero.
Similarly, integrating the isochoric pressure-temperature coefficient at constant volume (V) gives:
p = (p0/T0) T,
where p0 is the pressure at a reference temperature T0.
Combining the two relationships and eliminating the integration constants by considering that when T = T0, V = V0 and p = p0, we obtain the ideal gas law:
pV = RT,
which is the sought equation of state for the gas under consideration.