# Heat Capacity 101: Detecting and Measuring Heat Capacity Changes in Biological Reactions

Okay, after a long, long gap (on the blogosphere timescale) and/or almost zero elapsed time (by scientific literature standards), we're going to attempt to wrap up this mini-series on heat capacity effects in biology. Parts 1 and 2 are here and here, respectively.

So: How do you know if your reaction has a heat capacity change? Actually, it's easy: collect data as a function of temperature and make a van't Hoff plot (ln 1/Keq versus ln 1/temperture) or a Gibbs-Helmholtz plot (ÎG versus temperature) - if either plot is not-linear your reaction has a ÎCp. This figure shows Gibbs-Helmholtz plots (ÎG vs. temperature) and van't Hoff plots (ln K vs. 1/T) for the same reaction without (panels A and B) or with (panels C and D) a ÎCp. In all 4 plots, the ÎH, TÎS, and ÎG at 25 degrees C are the same. ÎH and ÎS do not vary with temperature in plots A and B, so both the Gibbs-Helmholtz (A) and van't Hoff (B) plots are linear. Panels C and D show the effect of including a ÎCp of -1.3 kcal/mol K.

Up until a couple of decades ago, it seemed that van't Hoff plots in biology were always linear. The slope of a van't Hoff plot gives you the enthalpy of the reaction (ÎH), and in actuality, if you only look a reaction over a relatively restricted temperature range, the van't Hoff plot often does look linear. But as people started working with extremophiles, and pushing experimental methods to work over wider temperature ranges, curved van't Hoff plots became more common. However, I occasionally get email from researchers who say that a colleague told them that they made a measurement mistake because they've got a curved van't Hoff plot. And I always tell them that curved van't Hoff plots are becoming the norm in biochemistry, as it turns out that a huge fraction of biological reactions have an associated heat capacity change. So, determining IF you have a ÎCp is relatively easy: just look at the plot and see if it is curved. A little more involved, but still relatively straightforward is to then determine the value of the ÎCp from the van't Hoff or Gibbs-Helmholtz plot.

If you have a linear van't Hoff plot, you can fit the data directly to the van't Hoff equation:

lnK = -ÎH/RT + ÎS/R

and the slope will provide you with the reaction enthalpy (ÎH). Linearity means that ÎCp for the reaction = 0, and that the ÎH value is the same at all temperatures.
The van't Hoff equation is actually also a rearrangement of the Gibbs-Helmholtz equation:

â(ÎG/T) / âT = -ÎH/T^2

The van't Hoff equation fits a line to a plot of lnK versus 1/T (with a slope of -ÎH/R), while the Gibbs-Helmholtz equation fits a line to a plot of ÎG versus T. When there is no ÎCp (i.e. when the plots are linear) it is far more common to use a van't Hoff plot and the van't Hoff equation than to use the Gibbs-Helmholtz.

However, if you have a non-linear van't Hoff, you need to perform a non-linear fit to either a "modified van't Hoff equation" or a "modified Gibbs-Helmholtz equation". With a curved plot, it is much more common to use the modified Gibbs-Helmholtz equation than to use a modified van't Hoff equation. Both are just expansions of the normal van't Hoff or Gibbs-Helmholtz equations that incorporate a temperature dependent enthalpy (i.e. a ÎH that changes with temperature):

ÎG(T) = ÎHr - TÎSr + ÎCp [T - Tr - T ln (T/Tr)]

It clear that this equation is a bit more complex than either of the linear forms. Also, since the enthalpy and entropy both change with temperature if there is a ÎCp, you need to pick a reference temperature to solve the equation (Tr), and then you get the enthalpy and entropy values at that reference temperature (ÎHr and ÎSr). Once you have your ÎCp, you can see if it follows one of the molecular correlations like those discussed in Part 2 of this "ÎCp in Biology 101" series. These methods of analysis are discussed in much more detail in a recent Methods in Enzymology article our lab just published (if anyone would like a reprint, just let me know).

It's all a bit esoteric, but at the same time, these relationships and plots, etc. are all the direct result of scientific advances in the 1800's that produced what we now know as the laws of thermodynamics, and formed the fundamental scientific underpinnings of the industrial revolution. If we can eventually understand how biological molecules utilize and control these same thermodynamic forces, then we can really begin to design true molecular engines and machines. It's still a long way off, but it means that perhaps one day we'll understand enzymes as well as we understand internal combustion engines.

Tags

### More like this

Very interesting post; would you be kind enough to send a PDF of your Met Enz article?

Best,
Mike

By Michael Marlow (not verified) on 13 Jul 2012 #permalink

I agree that is indeed an interesting post, and I would like to read the entire Met Enz article. Would you be so kind to send me a PDF?
With anticipated thanks and best regards,
Alice Mischie

By Alice Mischie (not verified) on 08 Nov 2013 #permalink

Hi I would love a copy of the full article if that is possible? best wishes
Bill

By William Stockham (not verified) on 18 Apr 2015 #permalink