Question

Calculate the molar kinetic energy (in joules per mole) of a sample of \(\mathrm{Ne}(\mathrm{g})\) at (a) \(25.00^{\circ} \mathrm{C}\) and (b) \(26.00^{\circ} \mathrm{C}\). (c) The difference between the answers to parts (a) and (b) is the energy per mole that it takes to raise the temperature of \(\mathrm{Ne}(\mathrm{g})\) by \(1^{\circ} \mathrm{C}\). The quantity is known as the molar heat capacity. What is its value?

Short answer

The molar kinetic energy at 25.00°C is 3706.1 J/mol, at 26.00°C it is 3722.0 J/mol, and the molar heat capacity (C_v) is 15.9 J/mol·K.

Step-by-step solution

Convert temperatures to Kelvin

Temperature in Kelvin (K) is found by adding 273.15 to the Celsius temperature. So, convert both temperatures from Celsius to Kelvin:(a) For 25.00°C: T(K) = 25.00 + 273.15 = 298.15 K(b) For 26.00°C: T(K) = 26.00 + 273.15 = 299.15 K

Calculate the molar kinetic energy for each temperature

The molar kinetic energy (E_k) for an ideal gas is given by the formula: E_k = (3/2)RT, where R is the universal gas constant (R = 8.3145 J/mol·K) and T is the temperature in Kelvin:(a) For T = 298.15 K: E_k(25°C) = (3/2) * R * T = (3/2) * 8.3145 J/mol·K * 298.15 K(b) For T = 299.15 K: E_k(26°C) = (3/2) * R * T = (3/2) * 8.3145 J/mol·K * 299.15 K

Solve for the molar kinetic energy at each temperature

Using the formula from Step 2, calculate the molar kinetic energies:(a) E_k(25°C) = (3/2) * 8.3145 * 298.15 J/mol(b) E_k(26°C) = (3/2) * 8.3145 * 299.15 J/mol

Compute the difference in molar kinetic energy

Subtract the molar kinetic energy at 25°C from the molar kinetic energy at 26°C to find the energy difference per mole for a 1°C increase in temperature:E_k difference = E_k(26°C) - E_k(25°C)

Determine the molar heat capacity

The difference in molar kinetic energy calculated in Step 4 corresponds to the molar heat capacity at constant volume for neon gas (C_v), because this difference represents the energy it takes to raise the temperature of 1 mole of gas by 1°C. Therefore, C_v = E_k difference.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental principle in chemistry and physics that relates the pressure, volume, and temperature of an ideal gas with the number of moles of gas present. The law is represented by the equation:
\(PV = nRT\)
Here, \(P\) stands for pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the universal gas constant, and \(T\) is the temperature in Kelvin. In this particular exercise, we assume neon gas behaves as an ideal gas, and we use the Ideal Gas Law to deduce the molar kinetic energy. The kinetic energy of a gas is related to temperature, an understanding of which is key in thermodynamics. Every particle in an ideal gas has an average kinetic energy that depends only on the temperature of the gas and not on its chemical identity or volume.
This law aids in understanding and predicting how gases will behave under different conditions and is essential to grasp before delving into more complex exercises like the one provided. Understanding that the molar kinetic energy can be calculated using the ideal gas law relationship is essential for this exercise.

Molar Heat Capacity

The molar heat capacity of a substance, often denoted as \(C_m\), is a physical property that indicates the amount of heat required to raise the temperature of one mole of the substance by one degree Celsius (or one Kelvin). There are two kinds of molar heat capacities: at constant volume \(C_v\) and at constant pressure \(C_p\).
In our exercise, when discussing the molar heat capacity, we are referring to the constant volume molar heat capacity. This is because the exercise assumes a closed system where neon gas does not expand or compress as its temperature changes. Therefore, the molar heat capacity relates directly to the change in molar kinetic energy for a change in temperature, as demonstrated in the difference between the energy calculations at 25°C and 26°C.

Importance of Molar Heat Capacity

Understanding the molar heat capacity is crucial for predicting how much energy is needed to change a substance’s temperature and is widely used in chemistry and engineering. For instance, in calorimetry, scientists use this property to determine the heat exchanges that occur during chemical reactions.

Temperature Conversion

Temperature conversion is a basic skill in science that allows us to translate the temperature from one scale to another. The most common scales are Celsius (°C), Fahrenheit (°F), and Kelvin (K). For scientific calculations, it is standard to use the Kelvin scale because it is an absolute temperature scale with zero at absolute zero - the point at which particles theoretically stop moving.
To convert from Celsius to Kelvin, the conversion formula used is:
\(T(K) = T(°C) + 273.15\).
In the exercise, this conversion is a key step, as the molar kinetic energy formula requires the temperature to be in Kelvin. The ability to convert temperatures accurately ensures that the calculations of the physical properties, such as kinetic energy or heat capacity, are correct.

Why Kelvin Matters

Using Kelvin, rather than Celsius, is important because it standardizes measurements across scientific disciplines and ensures that temperature-based calculations correlate directly with the energy particles of a substance.