Question

You are outside on a hot day, with the air temperature at \(T_{0}\). Your sports drink is at a temperature \(T_{\mathrm{d}}\) in a sealed plastic bottle. There are a few remaining ice cubes in the sports drink, which are at a temperature \(T_{\mathrm{i}}\), but they are melting fast. a) Write an inequality expressing the relationship among the three temperatures. b) Give reasonable values for the three temperatures in degrees Celsius.

Short answer

Question: On a hot day, a person decides to cool their sports drink using ice cubes. Considering the air temperature, the drink temperature, and the ice temperature, write an inequality expressing the relationship among the three temperatures and provide reasonable values for each temperature in degrees Celsius. Answer: The inequality representing the relationship among the air temperature (T₀), the drink temperature (T_d), and the ice temperature (T_i) is T_i < T_d < T₀. A possible set of temperatures in degrees Celsius could be T_i = 0°C, T_d = 10°C, and T₀ = 30°C.

Step-by-step solution

a) Write an inequality expressing the relationship among the three temperatures.

To find the relationship among the three temperatures, think about the physical situation: the air is hot, the drink is cooler than the air but warmer than the ice, and the ice cubes are melting. Since the ice is melting, it must be at a temperature lower than both the air and the sports drink. Therefore, we can write the inequality as: $$ T_i < T_d < T_0 $$ This means that the ice temperature is less than the drink temperature, which is, in turn, less than the air temperature.

b) Give reasonable values for the three temperatures in degrees Celsius.

Now, let's assign reasonable values to each temperature: - Ice temperature, \(T_i\): Since ice is melting, it must be at around its melting point, which is 0°C. - Sports drink temperature, \(T_d\): The drink is not as hot as the air but warmer than the ice cubes. A reasonable temperature could be 10°C; cold enough to still feel refreshing, but not too cold so that the ice would not melt. - Air temperature, \(T_0\): As it is a hot day, a reasonable temperature for the air could be 30°C. So, a possible set of temperatures is: $$ T_i = 0°C ,\, T_d = 10°C ,\, T_0 = 30°C $$

Key concepts

Temperature Gradient

In the study of thermodynamics, the temperature gradient is a crucial concept. It refers to the rate of change in temperature along a particular direction in space. Think of it like a slope in a graph, but instead of showing height changes, it represents temperature changes.
In our scenario, there is a clear temperature gradient between the air, sports drink, and ice. The air, at its hottest on a summer day, creates a gradient with the colder drink and ice inside it. This gradient drives the heat flow from warmer areas to cooler ones. Essentially, the energy in the form of heat is eager to move from where it is more concentrated, like in the 30°C air, towards less concentrated areas, like the 10°C drink, and finally towards the melting ice at 0°C.
Understanding the temperature gradient helps us predict how quickly heat might transfer between the air, drink, and ice. If the differences between temperatures are large, as in our case, the gradient is steep, which in turn means quicker heat transfer.

Heat Transfer

Heat transfer is the process of thermal energy moving from one object or substance to another. There are three main methods of heat transfer: conduction, convection, and radiation. In our given scenario, most heat transfer occurs due to conduction and convection.

• **Conduction**: This method occurs when objects are in direct contact. Here, the sports drink's bottle is touching both the warmer air and the ice cubes, acting as a medium for the heat to move from the air into the drink.
• **Convection**: This type of heat transfer happens when fluids (which can be liquids or gases) carry heat from one place to another. In our case, the warm air surrounds the bottle, moving heat through gentle air currents towards the cooler bottle.

Heat transfer tries to equalize temperature differences, leading to the melting of ice as heat moves from the 30°C air to the cooler drink, and ultimately to the 0°C ice. When studying heat transfer in thermodynamic systems, recognizing which method is at work can help predict how and where heat will move over time.

Phase Change

A phase change refers to the process where a substance changes from one state of matter to another. In the example of the sports drink, adding ice cubes results in a phase change from solid (ice) to liquid (water). This is due to the melting point of ice being exceeded during heat transfer.
While energy is continuously transferred from the hot air to the cooler ice, this energy is not raising the ice's temperature. Instead, it is used in breaking the bonds that hold the ice in a solid state. This process is known as **latent heat of fusion**. During a phase change, the temperature remains constant until the entire substance has transformed.

• When ice absorbs enough heat, it will change to its liquid state without any temperature rise, as all the energy goes into altering the molecular structure.
• This consumption of heat enables the cooling effect of the drink, as solid ice transitions to liquid water even while temperatures differ among the air, drink, and ice.

Understanding phase changes is crucial in thermodynamics because these transitions significantly affect how substances absorb or release energy.