Start out by solving your equation for dV/dP.chickymd said:Homework Statement
When pressure is applied to a liquid, its volume decreases. Assuming that the isothermal compressibility κ=-1/V(δV/δP) is independent of pressure, derive an expression for the volume as a function of pressure.
Homework Equations
The Attempt at a Solution
I don't know where to start, but I know the answer should be V2=V1 exp[-κ(P2-P1)].
Have you learned how to solve a differential equation like this by separation of variables or by integrating factor?chickymd said:δV/δP=-κV
I'm not sure what to do now.
Have you learned how to integrate dV/V?chickymd said:No, I haven't taken differential equations.
Do you know what the derivative of ln(x) with respect to x is equal to?chickymd said:No, I haven't.
OK. This is where we start.chickymd said:It's 1/x
chickymd said:-κV=δV/δP
-κδP=δV/V=δln(V)
I can get this far, but I don't know how to solve it now.
Excellent job! Just get rid of that extra equal sign in the last equation.chickymd said:Do I just integrate both sides from 1 to 2?
∫(from 1 to 2) -κδP=∫(from 1 to 2) δln(V)
-κ(P2-P1)=ln(V2)-ln(V1)=ln(V2/V1)
V2/V1=e^(-κ(P2-P1))
V2=V1=e^(-κ(P2-P1))
Is this correct?
The expression for volume as a function of pressure is V = k/P, where V is the volume and P is the pressure. The constant k depends on the specific conditions of the system, such as temperature and number of moles.
The expression for volume as a function of pressure is derived from the ideal gas law, which states that PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is the temperature. By rearranging this equation, we can derive the expression V = k/P.
The units of the constant k depend on the units used for pressure and volume. For example, if pressure is measured in atmospheres and volume in liters, then the units of k will be liters atmospheres. However, if pressure is measured in pascals and volume in cubic meters, then the units of k will be cubic meters pascals.
No, the expression for volume as a function of pressure is only valid for ideal gases. Real gases deviate from ideal behavior at high pressures and low temperatures, so the expression may not accurately predict the volume of a real gas under these conditions.
The expression for volume as a function of pressure is essentially the mathematical representation of Boyle's Law, which states that at a constant temperature, the volume of an ideal gas is inversely proportional to its pressure. This can be seen in the expression V = k/P, where as pressure increases, volume decreases and vice versa.