I need this to be done with steps a and b included, for a I got KT=(-1/Vm)*((2a/Vm^3)-(RT)/(Vm-b)2)-1 NOTICE THAT THIS IS A HUGE RECIPRICAL, if you do the algebra you should get:Vm2(Vm-b)2____________2a(Vm-b)2-RTVm3If that helps but in your walkthrough of part C please explain how RT will ever disappear, unless I am just missing a connection, even when you walk it through from the full partial of Vm wrt pressure at constant temperature... 3. Quantities such as the isothermal compressibility help us to determine changes in thethermodynamic state functions for "real substances". These quantities can be measuredthrough direct laboratory measurements, or through the use of substance-specificequations of state, such as the van der Waals equation.RTa2VmVm²(a) Work through the math to obtain a formula for the isothermal compressibility of a vander Waals gas.P =KT ==- b1 /0VmVm Op T[Hint: this will probably be significantly easier if you use the reciprocal relationship for partial derivatives!This way, you can work from the van der Waals equation as given above, with P as the dependentvariable. The resulting derivative is not necessarily “simple”, but you should be able to navigate throughwith some judicious use of the chain rule, and a bit of algebraic maneuvering.](b) Once you have obtained your result, demonstrate that the units work out correctly.(c) Next, demonstrate that your expression reduces to the ideal gas result (KT = 1) atlarge values of Vm.(d) Calculate the value of KT for a hypothetical ideal gas at T = 525 K and Vm =2.50 L mol-¹, and again at T = 525 K and Vm = 25.0 L mol-¹.(e) Calculate the value of KT for ethanol (as a van der Waals gas, use parameters from E & Rtable 7.4) when T = 525 K and Vm = 2.50 L mol-¹, and again at T = 525 K and Vm =25.0 L mol-¹.

To determine the value of isothermal compressibility factor kT for Van der Waal gas. kT =…

Answered: Quantities such as the isothermal…

Chemistry: The Molecular Science

5th Edition

ISBN:9781285199047

Author:John W. Moore, Conrad L. Stanitski

Publisher:John W. Moore, Conrad L. Stanitski

Chapter9: Liquids, Solids, And Materials

Section: Chapter Questions

Problem 45QRT: At the critical point for carbon dioxide, the substance is very far from being an ideal gas. Prove...

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I need this to be done with steps a and b included, for a I got K_T=(-1/Vm)*((2a/Vm^3)-(RT)/(Vm-b)²)^-1 NOTICE THAT THIS IS A HUGE RECIPRICAL, if you do the algebra you should get:

V_m²(V_m-b)^{2
____________}

2a(V_m-b)2-RTV_m³

If that helps but in your walkthrough of part C please explain how RT will ever disappear, unless I am just missing a connection, even when you walk it through from the full partial of Vm wrt pressure at constant temperature...

3. Quantities such as the isothermal compressibility help us to determine changes in the
thermodynamic state functions for "real substances". These quantities can be measured
through direct laboratory measurements, or through the use of substance-specific
equations of state, such as the van der Waals equation.
RT
a
2
Vm
Vm²
(a) Work through the math to obtain a formula for the isothermal compressibility of a van
der Waals gas.
P =
KT ==
- b
1 /0Vm
Vm Op T
[Hint: this will probably be significantly easier if you use the reciprocal relationship for partial derivatives!
This way, you can work from the van der Waals equation as given above, with P as the dependent
variable. The resulting derivative is not necessarily “simple”, but you should be able to navigate through
with some judicious use of the chain rule, and a bit of algebraic maneuvering.]
(b) Once you have obtained your result, demonstrate that the units work out correctly.
(c) Next, demonstrate that your expression reduces to the ideal gas result (KT = 1) at
large values of Vm.
(d) Calculate the value of KT for a hypothetical ideal gas at T = 525 K and Vm =
2.50 L mol-¹, and again at T = 525 K and Vm = 25.0 L mol-¹.
(e) Calculate the value of KT for ethanol (as a van der Waals gas, use parameters from E & R
table 7.4) when T = 525 K and Vm = 2.50 L mol-¹, and again at T = 525 K and Vm =
25.0 L mol-¹.

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