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7.3.4: Data Analysis

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    424809
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    Calculation of \(\Delta T_c\) from the thermogram using Logger Pro

    You will use Logger Pro software to analyze at least one thermogram collected during the experiment. Logger pro is available on computers in the P-chem lab, and is also available from Duke OIT. However, versions later than 3.8.6 have trouble importing the .txt data files from the calorimeter. If you would like version 3.8.6 for your own computer, it is available on the course Sakai site.

    The graph of temperature (T) versus time (t) produced in this experiment is called a thermogram and should resemble that shown in Figure \(\PageIndex{1}\). Because the initial (pre-period) and final (post period) slopes of the thermogram may be different, and because the temperature rise due to a chemical reaction is usually nonlinear, it is necessary to decide on a procedure to best estimate the true net temperature rise in the experiment. A common method is to extrapolate the post period slope backward in time and the pre-period slope forward in time (the dotted lines in Figure \(\PageIndex{1}\)).

    clipboard_e97592367c8f5c66134baf29ef35cd07c.png
    Figure \(\PageIndex{1}\): ypical Temperature Rise Curve for 6755 Calorimeter. (Duke Chem)

    The net temperature rise (\(\Delta T_c\)) can be measured from the difference of these two extrapolated lines at some intermediate time in the reaction period. The exact location of this intermediate time depends on the calorimeter, and on the reaction being studied. A frequently used criterion for estimating this time is to measure \(\Delta T_c\) at the point where the shaded areas in Figure 4 are equal. The temperature rise from the reaction increases exponentially with time, as \(T = T_i + A (1-e^{–kt})\), where \(A\) is the temperature rise (\(\Delta T_c\)) and \(k\) is a constant. It can be shown that these two shaded areas will be equal when \(t=\frac{1}{k}\), or when the temperature increase is approximately \(0.63\Delta T_c\). This procedure usually works well for chemical reactions of the type studied here. A procedure for graphical analysis for determination of \(\Delta T_c\) based upon this criterion is outlined below.

    1. Start Logger Pro by Clicking on the sjwq33ylZ8-TeEMTf9Jo0HMc4YGd6-TYoet-OR_zF3XTVsCpG7up4aG46hnxeDCk9GB-O9wnAuvwmSoZ7nqCXhA_mCEt6gyDLqeXtqrXs7ih1q_Rz_WGqWh8o2ux-EzVq1BpZMQWdzAgiIio7K3f6Y_eFG99Rqjhsi_7Ni9v_12knUEifi8I7V9fNvcKp4moqaZhb1_cNA icon on the computer main screen. When the program loads, you will see a Data Set window on the left side of the screen and a Graph window on the right. Click once in the Graph window and then select Graph Options from the Options menu. Be sure that Connect Points is not checked. The Point Protectors option should be checked.
    2. Select Import From – Text File from the File menu. When prompted, select your Windows Notepad (.txt) file for your first KCl run. The data will import directly into the Data Set window and the thermogram will appear in the Graph window.
      Fullscreen_1

    3. Look first at the Data Set window. Double Click on the Column heading above the first column of numbers. A dialog box will appear and you can change the name of the column from Column to Time. In the Units box enter Seconds. Repeat this procedure for Column 2 – changing it to Temperature. There is a Symbols entry box below the Units section where you will find the degree symbol so that you can enter oC for the units of the Temperature axis.
      DATA1

    4. Look now at the pre-period baseline. From within the Graph window, use the mouse to click and drag along a section of the pre-period baseline that accurately reflects the slope of this portion of the thermogram. The software will highlight the selected portion of the baseline. Click on the DATA1-1 button in the button bar at the top of the screen. A Linear Least-Squares line will be calculated for the selected data. The results will be shown in a text box inside the graph window. You can move the text box to any position in the graph window with your mouse. Double-click inside this text box and an Options box will appear. Put checks in the Slope and Intercept Standard Deviations boxes and click OK. Now when you look at the Least-Squares results you will also see the associated uncertainties in slope and intercept.
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    5. Repeat step 4 procedure for the post-period baseline.
    6. The Interpolate function in Logger Pro will be useful to find \(R\) and \(\Delta T_c\). To begin, Select Interpolate from the Analyze menu. A vertical bar running between the pre- and post-period Linear Least-Squares lines will now move with your cursor. In addition, a new text box appears which will show you the temperature of the pre- and post-period lines at the position of the bar.
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    7. Move the interpolation bar to a point close to the beginning of the reaction period (see the picture above and also Figure \(\PageIndex{1}\)). Calculate R at this point by subtracting the final interpolated temperature from the initial interpolated temperature. In the picture above \(R = 21.39938 – 22.73327 = – 0.33385^\circ C\). Multiply \(R\) by 0.63 and add this result to the initial interpolated temperature. In the example above: \[T_{0.63R}=21.73327+(0.63 (-0.33385))=21.52295^\circ C\]
    8. Move the Interpolation bar to the point where this temperature, T0.63R, intersects with the thermogram. You can follow the temperature rise of the data at the very lower-left of the Graph window.
      ls-lsfYmdm5aVx0CK8y-LEXs4vNL1-m_iJB1Wx4Db03zQNLcyq3fWYS-xncGMJDbGNYWkj-pCun-PO1_7rVokFSkZfVi55t5Ps2kfjB0xIMww6owFzbfIdoi6DLLta6l0cw24KKG_c55mFJ9oIFvnoSkoAeHk-i0OGyGodOXLlFcBIiVcDcaWchVndDktTUEf0wgb-HWfQ

    9. Read the initial temperature (\(T_i\)) and the final temperature (\(T_f\)) at the points of intersection of this line with the pre- and post-period lines and subtract these to determine the corrected temperature rise (\(\Delta T_c\):\[\Delta T_c = T_f – T_i\]
    10. Compare your calculated \(\Delta T_c\) with the temperature rise given in the calorimeter report for the run. They should be in agreement to within 1-2%. If there is a larger discrepancy, Make a note of this in your ELN. Discuss with your instructors to make sure you understand why this might happen.

    You have now seen how to analyze a thermogram using Logger Pro, to obtain good values of \(\Delta T_c\). For subsequent runs, choose either to use your calculated \(\Delta T_c\) from the manual data workup using LoggerPro, or the Temperature Rise from the calorimeter report. Even when the two do not agree within ~1-2% of each other, using one method consistently for all runs tends to give reasonable results.

    Calculation of the calorimeter heat capacity

    Use the calibration data from dissolution of KCl to calculate the heat capacity of your calorimeter (\(C_p\)). Each calorimeter has a unique \(C_p\). The \(C_p\) of your calorimeter should be a constant, provided you always use the same amount of water in each run and the temperature of the calorimeter components is constant. However, you can also calculate the heat capacity of the "empty" calorimeter (\(C_p'\)) without water.

    Dissolution of KCl in DI water absorbs 230.98 J/g at 25˚C. The quantity of heat absorbed on solution of KCl increases by 2.066 joules per gram for each degree drop in temperature below 25˚C. The total energy absorbed in the reaction is given by \[q = m[-230.98\frac{J}{g} - 2.066 \frac{J}{g ^\circ C}(25^\circ C-T_{0.63R})] \]

    where:

    • \(q =\) heat evolved in joules
    • \(m = \) mass of KCl in grams
    • \(T_{0.63R} = \) temperature at the \(0.63R\) point on the thermogram in \(^\circ C\).
    • The term, \(2.066 \frac{J}{g ^\circ C}(25-T_{0.63R})\) adjusts the heat of reaction to any temperature above or below the 25˚C reference temperature.

    To calculate \(C_p\) (in \(\frac{J}{^\circ C}\)) for your calorimeter and its contents, use the equation: \[C_p = \frac{q}{\Delta T_c}\]

    Determine the heat capacity of the empty calorimeter (\(C_p'\)) by subtracting the heat capacity 100.00 g of water from the \(C_p\). \[C_p' = C_p - (100.00 g \times C_w)\]

    where:

    \(C_p' = \) the heat capacity of the empty calorimeter in joules per \(^\circ C\)
    \(C_w = 4.1796 \frac{J}{g}\) is the specific heat capacity of water in joules per gram at \(25 ^\circ C\)
    \(100.00 g = \) the amount of water used (density of water at \(25 ^\circ C\) is 1 g/mL)

    Report the value of \(C_p'\) in your ELN.

    Calculate the heat of solution for each salt analyzed

    The enthalpy change for a reaction conducted at constant pressure is given by the following equation, as discussed in the Introduction section:

    \[\Delta {H}\left( {T}_1\right)=-\left[ {C}_{ {p}, {cal}}+ {C}_{ {p}, \text { products }}\right]\left( {T}_2- {T}_1\right) \label{10}\]

    From the equation above, the \(\Delta {H}\left( {T}_1\right)\) would be the change in enthalpy for the reaction. We can also calculate the change in enthalpy per mole of reactant using a modified form of Equation \ref{10}.

    \[\Delta {H}\left( {T}_1\right)=\frac{-\left[ {C}_{ {p}, {cal}}+ {C}_{ {p}, \text { products }}\right]\left( {T}_2- {T}_1\right)}{\text{moles of reactant (salt)}} \label{15}\]

    where: \[C_{p, \text { products }} = \text{volume of sol.} \times \text{density of sol.} \times \text{specific heat capacity of sol.} \]

    and,

    volume of sol. = total volume of reactant solution (100.00 mL, assume negligible volume change for dilute solution)

    density of sol. = 1.00 gram/mL (assumed for dilute aqueous solution)

    specific heat capacity of sol. = 4.18 joule/gram (assumed for dilute aqueous solution)

    Therefore, we can also express Equation \ref{15} as follows:

    \[\Delta {H}\left( {T}_1\right)=\frac{-\left[ {C}_{ {p}, {cal}}+ (\text{volume} \times \text{specific heat capacity of sol.})\right]\left( \Delta T_c\right)}{\text{moles of reactant (salt)}} \label{16}\]


    This page titled 7.3.4: Data Analysis is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Kathryn Haas.

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