3. 4. The coefficients of volumetric thermal expansion (p) and isothermal compre

Question

3.

4.    

The coefficients of volumetric thermal expansion (p) and isothermal compressibility (K) are defined by: beta = 1/v(delta V / delta T)p and K= -1/v (delta V/delta p)T Obtain expressions for Pideai and Kideai for a gas that obeys the ideal gas law. Simplify these expressions as much as possible and give the answers in terms of P and T only. Now derive expressions for P and k (as well as the ratios P/pi deal and K/Kideai) for a gas that obeys the van der Waals equation of state (P + )( - b) = RT, with = V/n. Simplify the final answers as much as possible. gives the expressions for coefficients of volumetric thermal expansion (Beta) and isothermal compressibility (K), which basically quantify how the volume of a substance changes with changing temperature and pressure, respectively. Using the expressions for (3 and K in Problem 3 and the fact that: (delta p/delta V)T(delta V/ delta T)P(delta T/delta P)V = -1 derive the expressions for (delta P/ delta T)V and (delta P/ delta V)T (which describe how the pressure of a substance changes with the temperature and volume) in terms of known material properties or quantities that are easily measurable in the lab such as (Beta, K, V, T, P, mldr Now, consider a vessel that is completely filled with liquid H2O and sealed at 25 degree C and 1 bar. What is the pressure of H2O inside the vessel if the temperature of the system is increased to 50 degree C. Under these conditions assume that Beta Water = 2.04 times 10-4 K-1, Beta vessel = 1.02 x 10-4 Hr1, and K water = 4.59 x 10-5 bar-1 and independent of temperature in the entire 25 to 50 degree C range being considered.

Explanation / Answer

solution 4a)

(dP/dV)T (dV/dT)P (dT/dP)v = -1      ------------- (1)

To get (dP/dT)

From (1)

(dP/dV)T (dV/dT)P = -1 / (dT/dP)v

(dP/dV)T (dV/dT)P = - (dP/ dT)

(dP/ dT) = - (dP/dV)T (dV/dT)P

(dP/dT) = - (dV/dT)P * (1 / (dV/dP)T)

(dP/dT ) = - (dV/dT) / (dV/dP)

multiply and divide by (1/V)

(dP/dT) = (1/V)(dV/dT) / [-(1/V)(dv/dP)]

(dP / dT) = beta / k

To get (dP/dV)

From (1)

(dP/dV)T = -1 / [(dV/dT)P (dT/dP)v]

(dP/dV) = -1 / (dV/dP)

Multiply and divide by 1/v

(dP/dV) = (1/V) / [-(1/V)(dV/dP)]

(dP/dV) = (1/V) / k

(dP / dV) = 1/ Vk

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