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17: Electrochemistry

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_(Ellgen)/17%3A_Electrochemistry

Thumbnail: Schematic of Zn-Cu galvanic cell. (CC BY-SA 3.0; Ohiostandard).

10.15: Problems

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_(Ellgen)/10%3A_Some_Mathematical_Consequences_of_the_Fundamental_Equation/10.15%3A_Problems

For

$S=S\left(P,V\right)$ , we obtain

${\left(\frac{\partial S}{\partial V}\right)}_P=\frac{C_P}{T}{\left(\frac{\partial T}{\partial V}\right)}_P \nonumber$ For

$S=S\left(P,T\right)$ , we obtain ...For

$S=S\left(P,V\right)$ , we obtain

${\left(\frac{\partial S}{\partial V}\right)}_P=\frac{C_P}{T}{\left(\frac{\partial T}{\partial V}\right)}_P \nonumber$ For

$S=S\left(P,T\right)$ , we obtain

${\left(\frac{\partial S}{\partial T}\right)}_P=\frac{C_P}{T} \nonumber$ For temperatures near 4 C and at a pressure of 1 atm, the molar volume of water is given by

$\overline{V}={\overline{V}}_4+a{\left(T-277.15\right)}^2 \nonumber$ where \({\overline{V}}_4=1.801575\times {10}^{-6}\ {\mathrm…

26.1: Appendix A. Standard Atomic Weights 1999†

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_(Ellgen)/26%3A_Appendices/26.01%3A_Appendix_A._Standard_Atomic_Weights_1999

\[ \begin{array}{l l l l} \text{Atomic Number} & \text{Name} & \text{Symbol} & \text{Atomic Weight} \\ \hline 1 & \text{Hydrogen} & \text{H} & 1.00794 \\ 2 & \text{Helium} & \text{He} & 4.002602 \\ 3 ...\[ \begin{array}{l l l l} \text{Atomic Number} & \text{Name} & \text{Symbol} & \text{Atomic Weight} \\ \hline 1 & \text{Hydrogen} & \text{H} & 1.00794 \\ 2 & \text{Helium} & \text{He} & 4.002602 \\ 3 & \text{Lithium} & \text{Li} & [6.941(2)] \\ 4 & \text{Beryllium} & \text{Be} & 9.012182 \\ 5 & \text{Boron} & \text{B} & 10.811 \\ 6 & \text{Carbon} & \text{C} & 12.0107 \\ 7 & \text{Nitrogen} & \text{N} & 14.0067 \\ 8 & \text{Oxygen} & \text{O} & 15.9994 \\ 9 & \text{Fluorine} & \text{F} & 18.998…

4.17: Problems

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_(Ellgen)/04%3A_The_Distribution_of_Gas_Velocities/4.17%3A_Problems

If all of the molecules that hit the hole escape, but the hole is so small that the number escaping has no effect on the velocity distribution of the remaining gas molecules, we call the escaping proc...If all of the molecules that hit the hole escape, but the hole is so small that the number escaping has no effect on the velocity distribution of the remaining gas molecules, we call the escaping process effusion. Third, the hole must be small enough so that the escaping molecules do not create a pressure gradient; the rate at which gas molecules hit the hole and escape must be determined entirely by the equilibrium distribution of gas velocities and, of course, the area of the hole.

13.4: The Gibbs Free Energy Change for Reaction at Constant Partial Pressures

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_(Ellgen)/13%3A_Equilibria_in_Reactions_of_Ideal_Gases/13.04%3A_The_Gibbs_Free_Energy_Change_for_Reaction_at_Constant_Partial_Pressures

When the ideal gases are separated from one another, the Gibbs free energy difference is the Gibbs free energy of

$c$ moles of gas

$C$ (at pressure

$P_C$ ) plus the Gibbs free energy of

$d$ mol...When the ideal gases are separated from one another, the Gibbs free energy difference is the Gibbs free energy of

$c$ moles of gas

$C$ (at pressure

$P_C$ ) plus the Gibbs free energy of

$d$ moles of gas

$D$ (at pressure

$P_D$ ) minus the Gibbs free energy of

$a$ moles of gas

$A$ (at pressure

$P_A$ ) and minus the Gibbs free energy of

$b$ moles of gas

$B$ (at pressure

$P_B$ ).

4.14: Collisions between like Gas Molecules

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_(Ellgen)/04%3A_The_Distribution_of_Gas_Velocities/4.14%3A_Collisions_between_like_Gas_Molecules

When we consider collisions between different gas molecules of the same substance, we can denote the relative velocity and the expected value of the relative velocity as

$v_{11}$ and \(\left\langle ...When we consider collisions between different gas molecules of the same substance, we can denote the relative velocity and the expected value of the relative velocity as

$v_{11}$ and

$\left\langle v_{11}\right\rangle$ , respectively. If we multiply the collision frequency per molecule,

${\widetilde{\nu }}_{11}$ , by the number of molecules available to undergo such collisions,

$N_1$ , we count each collision twice, because each such collision involves two type

$1$ molecules.

24: Indistinguishable Molecules - Statistical Thermodynamics of Ideal Gases

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_(Ellgen)/24%3A_Indistinguishable_Molecules_-_Statistical_Thermodynamics_of_Ideal_Gases

The ensemble analysis shows that the thermodynamic functions for an

$N$ -molecule system can be developed from the principles of statistical mechanics whether the molecules of the system interact or ...The ensemble analysis shows that the thermodynamic functions for an

$N$ -molecule system can be developed from the principles of statistical mechanics whether the molecules of the system interact or not. The theory is valid irrespective of the strengths of inter-molecular attractions and repulsions. In this chapter we apply the results from the ensemble theory to the particular case of ideal gases.

14.13: Relating the Differentials of Chemical Potential and Activity

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_(Ellgen)/14%3A_Chemical_Potential_-_Extending_the_Scope_of_the_Fundamental_Equation/14.13%3A_Relating_the_Differentials_of_Chemical_Potential_and_Activity

\[\begin{aligned} d{\mu }_A & = \left(\frac{\partial {\mu }_A}{\partial P}\right)_TdP+ \left(\frac{\partial {\mu }_A}{\partial T}\right)_PdT+ \left(d{\mu }_A\right)_{PT} \\ ~ & = \left(\frac{\partial ...\[\begin{aligned} d{\mu }_A & = \left(\frac{\partial {\mu }_A}{\partial P}\right)_TdP+ \left(\frac{\partial {\mu }_A}{\partial T}\right)_PdT+ \left(d{\mu }_A\right)_{PT} \\ ~ & = \left(\frac{\partial {\mu }_A}{\partial P}\right)_TdP+ \left(\frac{\partial {\mu }_A}{\partial T}\right)_PdT+\sum^{\omega }_{j=1} \left(\frac{\partial {\mu }_A}{\partial n_j}\right)_{PT}dn_j \\ ~ & =\overline{V}_AdP-\overline{S}_AdT+RT \left(d \ln \tilde{a}_A \right)_{PT} \\ ~ & =RT \left(\frac{\partial \ln \tilde{a}_A…

20.5: The Total Probability Sum at Constant N, V, and T

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_(Ellgen)/20%3A_Boltzmann_Statistics/20.05%3A_The_Total_Probability_Sum_at_Constant_N_V_and_T

So the total number of permutations,

$N!$ , over-counts the number of combinations by a factor of

$N_1!$ We can correct for this over-count by dividing by

$N_1!$ That is, after correcting for the...So the total number of permutations,

$N!$ , over-counts the number of combinations by a factor of

$N_1!$ We can correct for this over-count by dividing by

$N_1!$ That is, after correcting for the over-counting for the

$N_1$ molecules in the first energy level, the number of combinations is

${N!}/{N_1!}$ (If all

$N$ of the molecules were in the first energy level, there would be only one combination.

4.1: Distribution Functions for Gas-velocity Components

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_(Ellgen)/04%3A_The_Distribution_of_Gas_Velocities/4.01%3A_Distribution_Functions_for_Gas-velocity_Components

and

$v_x+dv_x$ ,

$y$ -components lie between

$v_y$ , and

$v_y+dv_y$ , and

$z$ -components lie between

$v_z$ and

$v_z+dv_z$ ,

$dP\left(\textrm{ʋ}\right)$ denotes the probability that the velo...and

$v_x+dv_x$ ,

$y$ -components lie between

$v_y$ , and

$v_y+dv_y$ , and

$z$ -components lie between

$v_z$ and

$v_z+dv_z$ ,

$dP\left(\textrm{ʋ}\right)$ denotes the probability that the velocity of a randomly chosen molecule,

$\ \left(v^*_x,v^*_y,v^*_z\right)$ , satisfies the conditions

$v_x<v^*_x<v_x+dv_x$ ,

$v_y<v^*_y<v_y+dv_y$ , and

$v_z<v^*_z<v_z+dv_z$ .

1.3: Classical Thermodynamics

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_(Ellgen)/01%3A_Introduction_-_Background_and_a_Look_Ahead/1.03%3A_Classical_Thermodynamics

If we confine an ideal gas in a frictionless piston and arrange to add heat to the gas while increasing the volume of the piston in a coordinated way, such that the temperature of the gas remains cons...If we confine an ideal gas in a frictionless piston and arrange to add heat to the gas while increasing the volume of the piston in a coordinated way, such that the temperature of the gas remains constant, the expanding piston will do work on some external entity, and the amount of this work will be just equal to the thermal energy added to the gas.

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