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Problem 84

$\mathrm{ABaSO}_{4}$ slurry is ingested before the gastrointestinal tract is $\mathrm{x}$ -rayed because it is opaque to $\mathrm{x}$ -rays and defines the contours of the tract. $\mathrm{Ba}^{2+}$ ion is toxic, but the compound is nearly insoluble. if $\Delta G^{\circ}$ at $37^{\circ} \mathrm{C}$ (body temperature) is 59.1 $\mathrm{kJ} / \mathrm{mol}$ for the process
$$\mathrm{BaSO}_{4}(s) \rightleftharpoons \mathrm{Ba}^{2+}(a q)+\mathrm{SO}_{4}^{2-}(a q)$$
what is $\left[\mathrm{Ba}^{2+}\right]$ in the intestinal tract? (Assume that the only source
of $\mathrm{SO}_{4}^{2-}$ is the ingested slurry.)

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## Video Transcript

were given this soluble ity reaction and its value for the changing Gibbs free energy. It's danger conditions at a temperature of 37 degrees Celsius, which one converted into Kelvin, comes out to 310 from this information. At these conditions, we want to determine the concentration of be a two plus ions in solution and from the information in the problem, we can conclude that the concentration of be a two plus is equal to the concentration of S 04 to minus and solution so we can begin by reading out the equilibrium constant expression where K equals when we look at the acquia species. Participating in this reaction equals B A two plus that concentration value won't applied by concentration of s 04 tu minus. Based on this relationship, we can see that this equals the concentration of be a two plus squared. And we also know an equation to relate the equilibrium constant with the change in Gibbs free energy value. So once we solve for that value for K, we can take the square root of it to isolate and find the concentration of barium ions in solution. So the equation, their release K in Delta G is K equals E to the power of negative Delta G over rt So K equals e to the power of negative Delta G, which were given as 59.1 village als per mole. But we should convert that into jewels promotion that we can cancel out the units of the constant are so we should multiply that by 1000. Get it from killing jewels to Jules per mole. And then we divide that by the constant are 8.314 joules per mole times Kelvin times that temperature of 310 Kelvin. And now when we cancel off the units of Kelvin and jewels Permal, we can successfully take e to the power of this quantity. Since it is dimension lis and we come up with a you mentioned this value for or K two B equal to concentration of B A two plus squared so that when we take the square root of this K that we found from this equation, we can get a concentration of barium ions. You be equal to 1.5 times 10 to the power of negative five bulls for leader