Problem 117 - General Chemistry: Principles and Modern Applications

Question

In the Are You Wondering \(7-1\) box, the temperature variation of enthalpy is discussed, and the equation \(q_{P}=\) heat capacity \(\times\) temperature change \(=C_{P} \times \Delta T\) was introduced to show how enthalpy changes with temperature for a constant-pressure process. Strictly speaking, the heat capacity of a substance at constant pressure is the slope of the line representing the variation of enthalpy (H) with temperature, that is $$C_{P}=\frac{d H}{d T} \quad(\text { at constant pressure })$$ where \(C_{P}\) is the heat capacity of the substance in question. Heat capacity is an extensive quantity and heat capacities are usually quoted as molar heat capacities \(C_{P, \mathrm{m}},\) the heat capacity of one mole of substance; an intensive property. The heat capacity at constant pressure is used to estimate the change in enthalpy due to a change in temperature. For infinitesimal changes in temperature, $$d H=C_{p} d T \quad(\text { at constant pressure })$$ To evaluate the change in enthalpy for a particular temperature change, from \(T_{1}\) to \(T_{2}\), we write $$\int_{H\left(T_{1}\right)}^{H\left(T_{2}\right)} d H=H\left(T_{2}\right)-H\left(T_{1}\right)=\int_{T_{1}}^{T_{2}} C_{P} d T$$ If we assume that \(C_{P}\) is independent of temperature, then we recover equation (7.5) $$\Delta H=C_{P} \times \Delta T$$ On the other hand, we often find that the heat capacity is a function of temperature; a convenient empirical expression is $$C_{P, \mathrm{m}}=a+b T+\frac{c}{T^{2}}$$ What is the change in molar enthalpy of \(\mathrm{N}_{2}\) when it is heated from \(25.0^{\circ} \mathrm{C}\) to \(100.0^{\circ} \mathrm{C} ?\) The molar heat capacity of nitrogen is given by$$C_{P, \mathrm{m}}=28.58+3.77 \times 10^{-3} T-\frac{0.5 \times 10^{5}}{T^{2}} \mathrm{JK}^{-1} \mathrm{mol}^{-1}$$

Step-by-Step Solution

Answer

The change in molar enthalpy of \( \mathrm{N}_{2} \) when it is heated from \( 25.0^{\circ} \mathrm{C} \) to \( 100.0^{\circ} \mathrm{C} \) can be calculated using the integral calculus. The exact answer cannot be given without performing the integration operation.

Step 1: Convert temperatures to Kelvin

To start with, convert the provided temperatures from Celsius to Kelvin. This is done by adding 273.15 to the Celsius temperature. Let \( T_1 = 25.0^{\circ}C = 298.15K \) and \( T_2 = 100.0^{\circ}C = 373.15K \).

Step 2: Define the molar heat capacity function

The given formula for molar heat capacity of nitrogen is \( C_{P,m} = 28.58 + 3.77 \times 10^{-3} T - \frac{0.5 \times 10^{5}}{T^2} JK^{-1}mol^{-1} \), where \( T \) is the temperature in Kelvin. This will be used later for finding the integral.

Step 3: Use Calculus to find ΔH

The ΔH from temperature \( T_1 \) to \( T_2 \) can be found by evaluating the integral \( \int_{T1}^{T2} C_{P, m} dT \). This integral will give us the molar change in enthalpy.

Step 4: Evaluate the integral

Now all that is left is to evaluate this integral. When this is done, it will give us the change in molar enthalpy for the given temperatures. Please note that evaluation of the integral requires knowledge of calculus and specific integral formulas.

Key Concepts

Heat Capacity

Heat capacity is a crucial concept in chemistry, especially when understanding how substances respond to temperature changes. It is defined as the amount of heat required to raise the temperature of a given amount of a substance by one degree Celsius (or Kelvin). In simpler terms, it's a measure of how much energy a substance can absorb before its temperature rises. There are two types of heat capacities:

• Specific Heat Capacity: Refers to the heat needed per unit mass of a substance.
• Molar Heat Capacity: Concerns the heat required per mole of a substance.

Heat capacity is an extensive property, meaning it depends on the amount of the substance present. This dependence makes heat capacity unique as it can vary drastically between different substances or different quantities of the same substance. For a constant-pressure process, the relationship between enthalpy and temperature is expressed by the heat capacity at constant pressure, denoted as \(C_P\). It plays a significant role when calculating enthalpy changes in processes involving heat transfer.

Molar Heat Capacity

Molar heat capacity is an extension of the concept of heat capacity but standardized per mole of substance, thereby remaining an intensive property. This means molar heat capacity does not depend on the amount of material but rather purely on the nature of the substance itself. When calculating the molar heat capacity, the formula often involves temperature-dependent expressions:

• Constant Value or Linear Form: May assume a constant value across a range or be represented as \(a + bT\).
• Complex Expressions: Can include terms like \(\frac{c}{T^2}\) to account for variability with temperature.

Let's consider \( N_2 \) nitrogen gas with its molar heat capacity given as: \[C_{P, \mathrm{m}} = 28.58 + 3.77 \times 10^{-3} T - \frac{0.5 \times 10^{5}}{T^{2}} \mathrm{J} \mathrm{K}^{-1} \mathrm{mol}^{-1}\] This equation implies that nitrogen's heat absorbing capacity changes nonlinearly with temperature, and understanding these changes is key to accurately determining enthalpy changes over temperature ranges.

Calculus in Chemistry

In chemistry, calculus is an indispensable tool, especially when dealing with changes that involve continuous variables like temperature and pressure. Calculus enables us to model the subtle overarching impacts of these variables on chemical processes mathematically. When evaluating enthalpy changes between two temperatures, calculus allows us to derive and integrate functions to find the total change over a range, rather than just at discrete points.

• Integrals help compute enthalpy changes precisely, as seen in our nitrogen example where we integrate its heat capacity: \(\int_{T_1}^{T_2} C_{P, \mathrm{m}} \, dT\).
• Differentials, such as \( \frac{dH}{dT} \), describe instantaneous rates of change and are crucial for deriving heat capacity functions.

By effectively using calculus, chemists can account for non-linear relationships and better predict how reactions and substances behave under various conditions, making it an essential component of thermodynamic study.