Asample containing 2.50 mol of an ideal gas at 298 k is expanded from an initial volume of 10.0 l to a final volume of 50.0 l. calculate δg and δa for this process for:
(a) an isothermal reversible path and (b) an isothermal expansion against a constant external pressure of 0.750 bar. explain why δg and δa do or do not differ from one another.

a) ΔG=ΔA= -33.452 J

b) ΔG=ΔA=0

Explanation:

a) according to the first law of thermodynamics the change in Gibbs free energy and Helmholtz free energy is

dG=VdP - SdT

dA=  -PdV - SdT

for an isothermal path dT=0 , therefore

ΔG=∫VdP = ∫(nRT/P)dP= nRT ln P2/P1

ΔA= ∫ (-P)dV = ∫ (-nRT/V)dV = nRT ln V1/V2

also from the ideal gas equation from an isothermal process

P1*V1= nRT = P2*V2 → P2/P1 = V1/V2

ΔG= nRT ln V1/V2 = 2.5 mol * 8.314 J/mol K P ln (10L/50L) = -33.452 J

for Helmholtz free energy is also

ΔA= nRT ln V1/V2 = -33.452 J

since the final energy is lower than the initial one , the process is spontaneous

for an isothermal process and isobaric process, dT=0 and dP=0, therefore

ΔG= VdP - SdT = 0

since → P2/P1 = V1/V2= constant → dV=0

ΔA=0

ΔG=ΔA for an ideal gas in an isothermal process since G and A differ only in the H and U terms , but since these ones depends only on temperature for an ideal gas (H(T)=U(T) + PV = U(T) + nRT) , they stay constant under an isothermal process and therefore ΔG=ΔA.

pre-sliced bread is a relatively recent development in baking technology. previous to that, it was always sold in a bakery in daily-made contiguous loaves with no wrapping, and was moderately perishable, usually being used with a day or two (which is why every bakery had a discounted “day-old” bread table.) therefore, pre-sliced bread would grow stale quickly. the hard crust provided a protective layer, and once the exposed dry slice left from the previous cut was removed, the interior was still soft.

what keeps bread soft is the moisture retained in the crumb (the light spongy part that’s not crust). the reason the crust develops is the surface of the dough gets exposed to hot dry air during baking which browns the outer layer and seals it, preventing all the moisture from escaping the dough. longer exposure to air will also eventually cause your bread to lose flavor and become stale.

people were used to cutting slices as they needed them and storing bread in a specially-made wooden or metal breadbox or pail with a lid to preserve it longer, but the first automatically sliced commercial loaves were produced on july 6, 1928, in chillicothe, missouri, using a machine invented by otto rohwedder, an iowa-born, missouri-based jeweler.[1] pre-sliced bread would indeed dry out, were it not for another 20th century technology: the plastic bag. bread could now be pre-sliced, and if the enclosing bag remains sealed as much as possible, the bread will remain soft and fresh for longer than if sitting unwrapped on a shelf or enclosed in a bread box that is not air-tight. sealed packaging meant bread could also be sold on store shelves and eliminate frequent trips to a bakery.

sealing bread in a plastic bag had another effect: the moisture in the loaf will re-distribute into the crust and soften it, which is sometimes more pleasing in a sandwich - especially for children whose teeth might not yet be strong enough to tear through a hard crust. a homemade loaf of bread can be sealed in a bag overnight and the crust will become soft like store-bought sandwich bread.

these were novel changes in how bread was produced and sold, which led to the coining of the phrase “the greatest thing since sliced bread.”