# Why could it be assumed that the standard enthalpy change is independent of temperature down to 25 °C?

For the reversible reaction of pyruvic acid to 2,2-dihydroxypropanoic acid, we calculated enthalpy using NMR and extrapolated from area (integration) and peaks etc.

We then plotted $$\ln K$$ vs $$1/T.$$ I am confused by this questions because I thought enthalpy was temperature dependent.

I guess my question was unclear. The exact question we are asked is:

With the assumption that the standard enthalpy change is independent of $$T$$ down to 25 °C, calculate an extrapolated value for $$K$$ (25 °C) and compare this to your experimental $$K$$ value determined at 25 °C. Is this a fair assumption? Why?

I do not understand if this assumption is fair and why?

• What "questions" are you confused about, exactly? It looks like you are doing an Arrhenius plot; why do you think there is an evidence for enthalpy being independent of $T?$ Nov 16 '19 at 12:38
• So the specific question we are asked is "With the assumption that the standard enthalpy change is independent of T down to 25°C, calculate an extrapolated value for K(25°C) and compare this to your experimental K value determined at 25°C. Is this a fair assumption? Why?" So we do do an arrhenius plot with points at temperatures 45, 50, 55, 60 and 65, and they we are supposed to extrapolate down to 25 to get a K value Nov 16 '19 at 12:41
• I removed the kinetics tag since the question does not involve kinetics. Nov 16 '19 at 13:44
• It is assumed that, over the temperature range of interest, the enthalpy change is approximately constant. Nov 16 '19 at 16:05
• I edited 2,2-dyhrydypropanic to 2,2-dihydroxypropanoic which is I guess what you meant, feel free to revert the post if that change is incorrect. Also, it would be nice if you avoided the ambiguous "etc" where it might not be self-evident what you did. Nov 16 '19 at 18:56

Generally speaking you can write that $$\Delta H^\circ(T) = \Delta H^\circ(T_\mathrm{ref}) + \int_{T_\mathrm{ref}}^T \Delta C_p \,\mathrm dT \tag{1}$$ where $$T_\mathrm{ref}$$ might be $$\pu{25 ^\circ C}$$ and the pressure is 1 bar (in your case, approximately 1 bar). If your temperature window is not too large you may get away with assuming $$\Delta H^\circ$$ is constant. This amounts to assuming that the second term in the equation, dependent on $$\Delta C_p$$, is small compared to the first. If it isn't then rather than using the integrated van't Hoff equation

$$\ln\left(\frac{K_2}{K_1}\right)=\frac{-\Delta H^\circ}{R}\left(\frac{1}{T_2}-\frac{1}{T_1}\right) \label{eqn:2}\tag{2}$$ maybe you should attempt a polynomial fit to $$\ln K_\mathrm{eq}$$ and taking the derivative of the polynomial wrt $$T^{-1}$$ to evaluate $$\Delta H^\circ$$ as a function of $$T$$.

To test whether \eqref{eqn:2} is appropriate you might evaluate the quality of a fit to that equation or check the linearity of a plot of $$\ln K$$ versus $$T^{-1}$$.

• The fit is 100% linear, the question is specifically about the validity of the assumption. I have the fit, and I have a value. Nov 16 '19 at 19:56
• Well, probably linearity is enough to justify extrapolation to the T of interest. Still, there is the question of how far you had to extrapolate. There are details you don't provide such as T points. Nov 16 '19 at 20:12
• For the high-temperature runs on sample 1, calculate K at each temperature from the integrated area values and plot In K versus the reciprocal of the absolute temperature. Determine the standard enthalpy change. With the assumption that the standard enthalpy change is independent of T down to 25°C, calculate an extrapolated value for K(25°C) and compare this to your experimental K value determined at 25°C. Is this a fair assumption? Why? Nov 16 '19 at 21:21
• I understand, but what does "high T" mean relative to 25C? Also, I suppose you might want to provide the Keq values at 25C (from the extrapolation and the "experimental" one you got either from a separate experiment or?) and from their difference (and uncertainty) you can answer the question "is this a fair assumption". The assumption is fair if the data supports it, in the absence of more info regarding heat capacities. You mention that the fits to vant Hoff are linear, which supports small $\Delta C_p$ so I'd say that alone justifies the assumption of constant $\Delta H$ Nov 16 '19 at 21:28
• So at 25 experimentally you get aroun 0.7, when you extrapolate you get 147.6, so several orders of magnitude difference. You can fit a 3rd order polynomial to the van;t hoff plot but I am not sure what this means Nov 16 '19 at 22:07