Question

At \(298 \mathrm{K}\) and 1 bar pressure, the density of water is \(0.9970 \mathrm{g} \mathrm{cm}^{-3},\) and \(C_{P, m}=75.3 \mathrm{JK}^{-1} \mathrm{mol}^{-1} .\) The change in volume with temperature is given by \(\Delta V=V_{\text {initial}} \beta \Delta T\) where \(\beta,\) the coefficient of thermal expansion, is \(2.07 \times 10^{-4} \mathrm{K}^{-1}\). If the temperature of \(325 \mathrm{g}\) of water is increased by \(25.5 \mathrm{K}\), calculate \(w, q, \Delta H,\) and \(\Delta U\)

Short answer

The values for work done (w), heat (q), change in enthalpy (ΔH), and change in internal energy (ΔU) for the given system are as follows: w = -171.86 J q = 34,522.25 J ΔH = 34,522.25 J ΔU = 34,694.11 J

Step-by-step solution

Calculate initial volume of water

Given the density of water is 0.997 g/cm³ and we have 325 g of water, we can calculate the initial volume as follows: Volume = Mass / Density V_initial = 325 g / 0.997 g/cm³ = 325.98 cm³ (approximately)

Calculate the final volume of water

To find the final volume, we will use the formula given for ΔV: ΔV = V_initial * β * ΔT Here: ΔV = Change in volume V_initial = 325.98 cm³ (from Step 1) β = 2.07 * 10^(-4) K^(-1) ΔT = 25.5 K ΔV = 325.98 cm³ * (2.07 * 10^(-4) K^(-1)) * 25.5 K = 1.7186 cm³ (approximately) The final volume, V_final, can be calculated as: V_final = V_initial + ΔV = 325.98 cm³ + 1.7186 cm³ = 327.6986 cm³ (approximately)

Calculate moles of water

Given that the molar mass of water is 18.015 g/mol, we can calculate the number of moles: n = Mass / Molar Mass n = 325 g / 18.015 g/mol = 18.049 moles (approximately)

Compute heat (q)

Now we can calculate the heat (q) using the formula q = nCpΔT: q = 18.049 moles * 75.3 J/(mol·K) * 25.5 K = 34,522.25 J (approximately)

Determine work done (w)

To find the work done, we will use the formula w = -PΔV. Given that the pressure is 1 bar: w = -1 bar * 1.7186 cm³ * (10^5 Pa/bar) * (10^-3 m³/cm³) = -171.86 J (approximately)

Calculate the change in enthalpy (ΔH)

The change in enthalpy (ΔH) can be found directly from the heat (q): ΔH = q = 34,522.25 J

Determine the change in internal energy (ΔU)

Now, we can calculate the change in internal energy (ΔU) using the formula: ΔU = ΔH - w = 34,522.25 J - (-171.86 J) = 34,694.11 J (approximately) Finally, we have the values for w, q, ΔH, and ΔU: w = -171.86 J q = 34,522.25 J ΔH = 34,522.25 J ΔU = 34,694.11 J

Key concepts

Heat Capacity

Heat capacity is an essential concept in thermodynamics and refers to the amount of heat required to change the temperature of a substance by a given amount. It is usually expressed in units of energy per temperature increment, such as joules per kelvin (J/K). The heat capacity can be further classified into two types: specific heat capacity and molar heat capacity.
Molar heat capacity, used in our exercise, is the heat required to raise the temperature of one mole of a substance by one degree Kelvin. For water in this problem, the molar heat capacity at constant pressure (\(C_{P,m}\)) is given as 75.3 J/K/mol. This value helps calculate the total heat absorbed by the water when its temperature is raised by 25.5 K.
Understanding heat capacity enables us to predict and calculate how substances will respond to changes in temperature, which is crucial in chemical reactions and various physical processes.

Enthalpy

Enthalpy, denoted by the symbol \(H\), is a thermodynamic property that represents the total heat content of a system. It is particularly useful in processes occurring at constant pressure, such as those taking place in an open container or the atmosphere. Enthalpy changes provide insights into the heat absorbed or released during chemical and physical transformations.
In our exercise, the change in enthalpy (\(\Delta H\)) is equivalent to the heat absorbed (\(q\)), as the process occurs at constant pressure. This is why we find \(\Delta H = 34,522.25\, J\). This information aids in understanding whether a process is endothermic (absorbing heat) or exothermic (releasing heat).

• Endothermic processes: \(\Delta H > 0\)
• Exothermic processes: \(\Delta H < 0\)

Enthalpy changes are pivotal for designing processes in chemistry, physics, and engineering.

Internal Energy

Internal energy, denoted by \(U\), is the total energy contained within a chemical system. It encompasses both kinetic energy (due to the movement of particles) and potential energy (due to interactions between particles). Internal energy is a key concept for understanding energy changes in chemical reactions and phase changes.
The change in internal energy (\(\Delta U\)) of a system is calculated from the first law of thermodynamics:\\[\Delta U = \Delta H - w\]where \(w\) is the work done by or on the system. In this exercise, given the enthalpy change and calculated work done (\(w = -171.86\, J\)), we find \(\Delta U = 34,694.11\, J\).
Understanding \(\Delta U\) helps determine the net energy change within a system and accounts for both heat and work interactions. It is useful when analyzing the overall efficiency and energy conservation in thermodynamic processes.

Thermal Expansion

Thermal expansion describes the tendency of materials to change in volume in response to changes in temperature. When substances are heated, their molecules move more vigorously, leading to an expansion in volume. This concept is characterized by the coefficient of thermal expansion (\(\beta\)), which quantifies how much a material expands per degree of temperature increase.
In the exercise, we calculate the change in volume (\(\Delta V\)) using the formula:\[\Delta V = V_{\text{initial}} \times \beta \times \Delta T\]With \(\beta = 2.07 \times 10^{-4}\, K^{-1} \), the expansion of water was found to be small, given that water has a low thermal expansion coefficient.

• Substances with high \(\beta\): Expand significantly with temperature changes.
• Substances with low \(\beta\): Have minimal volume changes.

Understanding thermal expansion is vital for applications in construction, manufacturing, and material science, as changes in volume could affect structural integrity and function.