To prepare nuclear fuel, U3 O8( "yellow cake" ) is converted to UO2(NO3)2, which is then conver

To prepare nuclear fuel, U3 O8( "yellow cake" ) is converted...

To prepare nuclear fuel, $\mathrm{U}_{3} \mathrm{O}_{8}(\text { "yellow cake" ) is converted }$ to $\mathrm{UO}_{2}\left(\mathrm{NO}_{3}\right)_{2}$, which is then converted to $\mathrm{UO}_{3}$ and finally UO2. The fuel is enriched (the proportion of the 235U is increased) by a two-step conversion of $\mathrm{UO}_{2}$ into$\mathrm{UF}_{6}$ , a volatile solid, followed by a gaseous-diffusion separation of the 235U and 238U isotopes: $$\begin{aligned} \mathrm{UO}_{2}(s)+4 \mathrm{HF}(g) & \longrightarrow \mathrm{UF}_{4}(s)+2 \mathrm{H}_{2} \mathrm{O}(g) \\ \mathrm{UF}_{4}(s)+\mathrm{F}_{2}(g) & \longrightarrow \mathrm{UF}_{6}(s) \end{aligned}$$ Calculate $\Delta G^{\circ}$ for the overall process at $85^{\circ} \mathrm{C} :$

To prepare nuclear fuel, $\mathrm{U}_{3} \mathrm{O}_{8}(\text { "yellow cake" ) is converted }$ to $\mathrm{UO}_{2}\left(\mathrm{NO}_{3}\right)_{2}$, which is then converted to $\mathrm{UO}_{3}$ and finally UO2. The fuel is enriched (the proportion of the 235U is increased) by a two-step conversion of $\mathrm{UO}_{2}$ into$\mathrm{UF}_{6}$ , a volatile solid, followed by a gaseous-diffusion separation of the 235U and 238U isotopes:
$$\begin{aligned} \mathrm{UO}_{2}(s)+4 \mathrm{HF}(g) & \longrightarrow \mathrm{UF}_{4}(s)+2 \mathrm{H}_{2} \mathrm{O}(g) \\ \mathrm{UF}_{4}(s)+\mathrm{F}_{2}(g) & \longrightarrow \mathrm{UF}_{6}(s) \end{aligned}$$
Calculate $\Delta G^{\circ}$ for the overall process at $85^{\circ} \mathrm{C} :$

Key Concepts

Gibbs Free Energy

Gibbs free energy (?G) is a thermodynamic function that combines enthalpy and entropy to predict the spontaneity of a process. A negative ?G indicates a spontaneous reaction under the given conditions, while a positive ?G signifies non-spontaneity.

Standard Free Energy Change Calculation

The standard free energy change for a reaction is determined by using the standard free energies of formation for the reactants and products. In a multi-step process, the overall ?G° is obtained by summing the ?G° values for each individual reaction step, enabling the evaluation of the net energetic favorability of a complex process.

Temperature Dependence in Thermodynamics

Since Gibbs free energy is temperature-dependent, calculations performed at non-standard temperatures (i.e., temperatures differing from 25°C or 298 K) require adjustments. This involves converting the temperature to Kelvin and considering how both enthalpic and entropic contributions change with temperature.

Multi-step Reaction Processes

In processes that involve multiple sequential reactions, the net change in free energy is the algebraic sum of the individual free energy changes. This principle of additivity is critical when analyzing complex reaction schemes, ensuring that each reaction's contribution to the overall energy change is appropriately accounted for.

Transcript

00:01 We are given two intermediate reactions for the overall reaction used to prepare nuclear fuel.

00:08 We want to determine what the change in gibbs free energy is of the overall reaction at 85 degrees celsius.

00:16 So the first thing that we need to do is combine these two reactions by adding them together in order to first write out that overall reaction.

00:25 We can see that uf4 cancels out on each side, and so the overall reaction is uo2 solid plus 4hf gas plus f2 gas goes to uf6 solid plus 2h2 gas.

01:03 And now we need to determine the delta g of this reaction at 85 degrees celsius.

01:14 Because the temperature is not at standard conditions of 298 kelvin, we cannot directly solve for delta g by finding the overall change in gibbs -free energy of the products and subtracting that same value for the reactants.

01:29 Instead, we need to use this equation since it relates temperature to delta g, through delta h and delta s and we are given values in the problem for delta h and delta s and we can use the appendix in order to look up those values for the ones that aren't given in the problem statement so these are the two equations that we are going to use to find delta h of reaction and delta s of reaction so that once we find them we can plug them into this equation at the temperature of 85 degrees celsius to find delta g at that temperature.

02:09 So for delta h, we first find the total change in enthalpy of the products and then subtract the total change in enthalpy of the reactants.

02:19 For the change in entropy, delta s of the reaction, we find the total entropy of the products and subtract the total entropy of the reactants.

02:30 So starting with delta h, we can break up, the change in enthalpy of the products and reactants into two different expressions to help us organize our work.

02:43 We start with the reactants.

02:46 Again, we are going off of this overall equation now.

02:51 We start with the products and we see that we have one mole of uf6 and two moles of h2o on the product side.

02:58 So one mole of uf6 and two moles of h2o, we are given this value for the delta h of uf6, we can look up this value for the delta h of h2o and the appendix.

03:11 And again, these correspond to the standard change in enthalpy of formation of each one of those compounds.

03:20 And so when we cancel off units of moles, we are left with energy units of kilojoules for the change in enthalpy of the products.

03:30 And since we added the total enthalpy change of each one of the products and add them together, that is the total enthalpy change of all of the products.

03:39 And now for the reactants, we have one mole of uo2, four moles of hf, and one mole of f2.

03:48 But remember that for delta h values, we use delta h of formation at standard conditions in order to help us determine the overall delta h.

03:58 And f2 is a naturally occurring diatomic gas molecule that has a delta h of formation value of zero.

04:06 So we only include the reactants of 1 mole of uo2 and 4 moles of hf for the delta h for the reactants...

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