What is the relationship between enthalpy and internal energy? Where does #q = DeltaH# come from?

Answer 1

SUBTLE ASSUMPTIONS

All right, let's examine that formula. You state that:

#q_p = DeltaE + PDeltaV = DeltaH#

This is the result of your formula:

#q_p = DeltaE + PDeltaV = DeltaH#
#q_p = q + w + PDeltaV = DeltaH#
#q_p = q - cancel(PDeltaV) + cancel(PDeltaV) = DeltaH#
#color(green)(q_p = DeltaH)#
Here, you have implicitly assumed ahead of time that #DeltaP = 0# and thus #VDeltaP = DeltaPDeltaV = 0#. In this case, #DeltaH = q_p#, so this formula is correct.

However, it hides the presumption that the pressure is constant—one that you may not even be aware you were making.

HEAT FLOW VS. ENTHALPY

Let's see what that means by comparing it to two related equations and a formula that I know to be accurate (and from which I would begin an enthalpy derivation of this type):

#\mathbf(DeltaH = DeltaE + Delta(PV))#
#DeltaE = q + w#
#w = -PDeltaV#

When determining relationships between internal energy and enthalpy under various conditions, the bolded equation is more helpful since it displays the entire relationship before any assumptions are made.

Next, you should notice that #Delta(PV)# requires you to use the product rule (plus a bit extra), which means you involve the change in volume and the change in pressure like so:
#Delta(PV) = PDeltaV + VDeltaP + DeltaPDeltaV#

Consequently, the enthalpy vs. heat flow relationship is:

#DeltaH = q + w + PDeltaV + VDeltaP + DeltaPDeltaV#
#= q - cancel(PDeltaV) + cancel(PDeltaV) + VDeltaP + DeltaPDeltaV#
#color(blue)(DeltaH = q + VDeltaP + DeltaPDeltaV)#

Enthalpy, then, is the product of heat flow, pressure variations at a given initial volume, and the simultaneous effects of both changes.

At this point, it is obvious that a constant pressure yields #DeltaH = q = q_p#.

INTERNAL ENERGY VS. ENTHALPY

Now, we can make comparisons between enthalpy and internal energy (recall the first law of thermodynamics for #DeltaE#):
#color(blue)(DeltaH = q + VDeltaP + DeltaPDeltaV)#
#color(blue)(DeltaE = q - PDeltaV)#

From this, we can deduce the following:

You should see that the heat flow is the only factor affecting enthalpy at constant pressure.

You should also note that the heat flow is the only factor that determines internal energy at a constant volume.

Ultimately, the equation in bold should show us that:

Enthalpy is the sum of internal energy and any labor needed to alter one of the two variables without considering the other.

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Answer 2

Enthalpy is defined as the sum of internal energy and the product of pressure and volume. Mathematically, H = U + PV. When a reaction occurs at constant pressure, the change in enthalpy, ΔH, is equal to the heat transferred, q, at constant pressure. Therefore, q = ΔH. This relationship stems from the definition of enthalpy and the first law of thermodynamics, which states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system on its surroundings, ΔU = q - W. At constant pressure, the work done is given by W = -PΔV, where P is the pressure and ΔV is the change in volume. Substituting this into the first law equation and rearranging yields ΔH = ΔU + PΔV, which simplifies to q = ΔH at constant pressure.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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