Question

The following data are a DSC scan of a solution of a T4 lysozyme mutant. From the data determine \(T_{m}\). Determine also the excess heat capacity \(\Delta C_{P}\) at \(T=308 \mathrm{K}\). Determine also the intrinsic \(\delta C_{P}^{i n t}\) and transition \(\delta C_{P}^{t r s}\) excess heat capacities at \(T=308 \mathrm{K}\). In your calculations use the extrapolated curves, shown as dotted lines in the DSC scan.

Short answer

To find the melting temperature \(T_{m}\), locate the point where the transition occurs; the temperature where the excess heat capacity reaches its peak on the DSC curve. To determine the excess heat capacity \(\Delta C_{P}\) at \(T = 308 \mathrm{K}\), find the values of the intrinsic \(\delta C_{P}^{int}\) and transition \(\delta C_{P}^{trs}\) curves at \(T = 308 \mathrm{K}\) using the graph and interpolated curve equation, then calculate the difference between these values. Lastly, obtain the values for \(\delta C_{P}^{int}\) and \(\delta C_{P}^{trs}\) at \(T = 308 \mathrm{K}\) from the respective extrapolated curves or using the interpolated equations.

Step-by-step solution

Locate \(T_{m}\)

To determine the melting temperature \(T_{m}\), carefully examine the DSC scan. We need to locate the point where the transition occurs. In this case, \(T_{m}\) (in K) will correspond to the temperature at which the excess heat capacity reaches its peak, or the highest point on the DSC curve. Use any specific point finding methods on the software of your choice to find this maximum point.

Determine \(\Delta C_{P}\) at \(T = 308 \mathrm{K}\)

The excess heat capacity \(\Delta C_{P}\) is the difference between the extrapolated curves at a specific temperature (in this exercise, \(T=308 \mathrm{K}\)). Use the graph and interpolated curve equation to find the values of heat capacity for both the intrinsic \(\delta C_{P}^{int}\) and transition \(\delta C_{P}^{trs}\) curves at \(T = 308 \mathrm{K}\). Now, calculate the difference between these values to find the excess heat capacity: \[\Delta C_{P} = \delta C_{P}^{int}(308) - \delta C_{P}^{trs}(308)\]

Determine \(\delta C_{P}^{int}\) and \(\delta C_{P}^{trs}\) at \(T = 308 \mathrm{K}\)

You now need to obtain the values for \(\delta C_{P}^{int}\) and \(\delta C_{P}^{trs}\) at 308K, using the plot and the interpolated equations. Look for the values on the respective extrapolated curves (intrinsic \(\delta C_{P}^{int}\) and transition \(\delta C_{P}^{trs}\)) at the point where \(T=308 \mathrm{K}\). Once you've determined these two values from the graph or obtained them from the interpolated equations, you have completed the exercise and obtained all the necessary parameters.

Key concepts

Melting Temperature

In the context of differential scanning calorimetry (DSC), the melting temperature, symbolized as \(T_m\), is crucial for understanding how a substance transitions from solid to liquid. This particular temperature signifies the point at which a material has absorbed enough heat energy to change its state.
This transition is marked by a notable peak on the DSC curve, which occurs due to the absorption of heat as the substance melts. The height and shape of this peak are indicators of the purity and composition of the sample being analyzed.

• To locate \(T_m\), check the graph from the DSC scan and identify where the peak occurs.
• The highest point on the DSC curve represents the melting temperature.
• This peak indicates that the maximum amount of energy is being absorbed at these conditions.

Observing \(T_m\) provides insights into the thermal properties of proteins, such as lysozymes, and can be compared across different substances or conditions.

Excess Heat Capacity

Excess heat capacity, denoted as \(\Delta C_P\), reflects changes in a substance's heat capacity during a transition, like melting. When observing a DSC scan, excess heat capacity is identified by the difference between the sample's intrinsic heat capacity and transition heat capacity at any given temperature.
This concept is essential for understanding energy absorption beyond the regular pattern due to transitions like phase changes.

• In the case of the exercise, use the extrapolated data from the DSC scan.
• Determine \(\Delta C_P\) by finding the difference between the point on the intrinsic heat capacity curve and the transition heat capacity curve at \(T=308 \text{ K}\).

This calculation tells us how much extra energy a sample requires at a specific temperature during phase change."

Intrinsic Heat Capacity

Intrinsic heat capacity \(\delta C_{P}^{int}\) is a fundamental characteristic of a material, signifying its ability to absorb heat without undergoing a phase transition.
In the context of DSC, it reflects the material's heat capacity under standard conditions without the influence of a transition such as melting or boiling.

• To obtain \(\delta C_{P}^{int}\) at a given temperature (e.g., \(308 \text{ K}\)), examine the extrapolated graph where no transition is occurring.
• This is the baseline or normal heat capacity the material exhibits.
• Typically used as a reference point in calculations.

By understanding \(\delta C_{P}^{int}\), researchers can ascertain how much energy a material can naturally absorb until it reaches a transition.

Transition Heat Capacity

Transition heat capacity \(\delta C_{P}^{trs}\) represents the heat capacity directly associated with a phase change. It consists of the additional heat capacity observed as the material undergoes a transition from one phase to another.
This heat capacity is explicitly related to the energy changes occurring during a transformation process.

• To find \(\delta C_{P}^{trs}\) at a particular temperature, observe the enhanced area under the DSC curve where the transition is occurring.
• This value is typically higher than \(\delta C_{P}^{int}\) due to the energy required for the phase change.
• Understanding \(\delta C_{P}^{trs}\) helps in quantifying the amount of energy absorbed specifically due to the transition.

With this concept, scientists can analyze a material's behavior during phase transformations, offering a window into its structural and thermal properties.