Question

It has been suggested that the surface melting of ice plays a role in enabling speed skaters to achieve peak performance. Carry out the following calculation to test this hypothesis. At 1 atm pressure, ice melts at \(273.15 \mathrm{K}\) \(\Delta H_{f u s i o n}=6010 . \mathrm{Jmol}^{-1},\) the density of ice is \(920 . \mathrm{kg} \mathrm{m}^{-3}\), and the density of liquid water is \(997 . \mathrm{kg} \mathrm{m}^{-3}\). a. What pressure is required to lower the melting temperature by \(4.0^{\circ} \mathrm{C} ?\) b. Assume that the width of the skate in contact with the ice has been reduced by sharpening to \(19 \times 10^{-3} \mathrm{cm}\) and that the length of the contact area is \(18 \mathrm{cm} .\) If a skater of mass \(78 \mathrm{kg}\) is balanced on one skate, what pressure is exerted at the interface of the skate and the ice? c. What is the melting point of ice under this pressure? d. If the temperature of the ice is \(-4.0^{\circ} \mathrm{C},\) do you expect melting of the ice at the ice-skate interface to occur?

Short answer

In short, we calculated the pressure required to lower the melting temperature of ice by $4.0^\circ C$ using the Clapeyron equation and found the pressure exerted by the skater at the ice-skate interface. We then determined the melting point of ice under this pressure and compared it to the given ice temperature of $-4.0^\circ C$. If the melting point is less than or equal to the given temperature, melting will occur at the ice-skate interface.

Step-by-step solution

(Step 1: Find the pressure required to lower the melting temperature)

We will use the Clapeyron equation for phase transitions of a substance: \[P = \dfrac{\Delta H_\mathrm{fusion}}{T\Delta V}\] Here, \(P\) represents the pressure difference required to change the melting temperature, \(\Delta H_{\mathrm{fusion}}\) is the enthalpy of fusion, \(T\) is the temperature difference, and \(\Delta V\) is the change in volume per mole. Using the given densities of ice and liquid water, as well as the molar volume, we can calculate \(\Delta V\) with the following formula: \[\Delta V = \rho_\mathrm{ice}^{-1} - \rho_\mathrm{water}^{-1}\] Then, we can find the required pressure difference to lower the melting temperature by 4.0°C.

(Step 2: Calculate pressure exerted by the skater)

To calculate the pressure exerted by the skater at the ice-skate interface, we will use the following equation: \[P = \dfrac{F}{A}\] Here, \(P\) represents the pressure being exerted, \(F\) is the force exerted (which is equal to the weight of the skater) and \(A\) is the area of the ice-skate interface. The weight of the skater can be found using the equation \(F = mg\), where \(m\) is the mass of the skater and \(g\) is the acceleration due to gravity (approximately \(9.81 \mathrm{m/s^2}\)). We will then calculate the area of the ice-skate interface using the given width and length.

(Step 3: Find the melting point of ice under the pressure exerted by the skater)

We will use the Clapeyron equation again: \[P = \dfrac{\Delta T \Delta H_\mathrm{fusion}}{T\Delta V}\] Here, \(P\) is the pressure exerted by the skater, and \(\Delta T\) is the temperature difference between the initial melting point and the melting point under this pressure. We can rearrange this equation to solve for \(\Delta T\), then find the new melting point by adding the initial melting point to \(\Delta T\).

(Step 4: Determine if melting occurs at the ice-skate interface)

Compare the melting point of ice under the exerted pressure with the given temperature of the ice (-4.0°C). If the melting point is less than or equal to the given temperature, melting will occur at the ice-skate interface.

Key concepts

Clapeyron Equation

The Clapeyron equation is a powerful tool used to describe the relationship between pressure and temperature during phase transitions, such as melting or boiling. It defines how much the pressure must change to achieve a certain change in temperature at the phase boundary. This equation is relevant when discussing melting ice under the blades of speed skaters, as it helps to calculate the pressure needed to lower the melting point of ice—a principle that may be behind the smooth glide of a skate.

When applying the Clapeyron equation, \( P = \frac{\Delta H_{\mathrm{fusion}}}{T\Delta V} \), where \( P \) is the pressure difference, \( \Delta H_{\mathrm{fusion}} \) is the enthalpy of fusion, \( T \) is the initial melting temperature in Kelvin, and \( \Delta V \) is the change in volume during fusion, you can explore how pressure affects melting points. It’s a principle not just limited to sports but also relevant in various scientific and industrial processes, like how pressure affects the cooking time in pressure cookers.

Enthalpy of Fusion

Enthalpy of fusion, denoted as \( \Delta H_{\mathrm{fusion}} \), is the amount of energy needed to change a substance from a solid to a liquid at its melting point without changing its temperature. This key thermodynamic quantity is essential for calculating the energy required during the phase transition and is a core part of understanding the surface melting of ice. For example, the enthalpy of fusion for ice is 6010 J/mol, which indicates how much energy is needed to melt one mole of ice at its melting point. This value can then be plugged into the aforementioned Clapeyron equation, establishing a direct link between the energy involved in the phase transition and the pressure required to induce melting at different temperatures.

Melting Point Depression

Melting point depression is a phenomenon where the melting point of a material decreases when pressure is applied. This is particularly relevant for understanding how ice melts beneath a speed skater's blade. According to the Clapeyron equation, increasing pressure leads to a lower melting temperature, allowing ice to melt at temperatures below its typical melting point at 1 atm pressure. This concept is not only applicable to the physics of skating but also in the culinary arts, such as when a chef makes ice cream, where salt is added to ice to lower its melting point and create a colder environment.

Pressure-Temperature Relationship

In the context of phase transitions, the pressure-temperature relationship describes how the phase of a material can change from solid to liquid (or liquid to gas) not just by changing the temperature but also by altering the pressure. For instance, the pressure exerted by a skater's blade on ice can significantly increase the temperature at which melting occurs. The Clapeyron equation encapsulates this relationship, making it a fundamental concept to grasp for students studying thermodynamics, meteorology, or material science, where understanding how temperature and pressure interact is crucial.

Phase Transition

A phase transition is the transformation of a substance from one state of matter to another, such as solid ice turning into liquid water. This is a central concept in thermodynamics and materials science, playing a key role in many natural and engineered processes. Ice melting under a skate blade is a prime example of a phase transition facilitated by the pressure of the blade. Phase transitions are not always just about solid, liquid, and gas; they also include transitions to states like plasmas or Bose-Einstein condensates, showcasing the wide-ranging implications of this foundational scientific principle.