Problem 53 - Calculus for Biology and Medicine

Question

To measure temperature, three scales are commonly used: Fahrenheit, Celsius, and Kelvin. These scales are linearly related. (a) The Kelvin (K) scale is an absolute scale of temperature. The zero point of the scale \((0 \mathrm{~K})\) denotes absolute zero, the coldest possible temperature; that is, no body can have a temperature below \(0 \mathrm{~K}\). It has been determined experimentally that \(0 \mathrm{~K}\) corresponds to \(-273.15^{\circ} \mathrm{C}\). If \(1 \mathrm{~K}\) denotes the same temperature difference as \(1^{\circ} \mathrm{C}\), find an equation that relates the Kelvin and Celsius scales. (b) Pure nitrogen and pure oxygen can be produced cheaply by first liquefying purified air and then allowing the temperature of the liquid air to rise slowly. Since nitrogen and oxygen have different boiling points, they are distilled at different temperatures. The boiling point of nitrogen is \(77.4 \mathrm{~K}\) and of oxygen is \(90.2 \mathrm{~K}\). Convert each of these boiling-point temperatures into Celsius. If you solved Problem \(52(\) a), convert the boiling-point temperatures into Fahrenheit as well. Consider the two techniques described for distilling nitrogen and oxygen. Which element gets distilled first?

Step-by-Step Solution

Answer

Equation for K and °C: \( T(K) = T(°C) + 273.15 \). Nitrogen distills first due to lower boiling point.

Step 1: Equation Relating Kelvin and Celsius

To find an equation relating the Kelvin (K) and Celsius (°C) scales, use the information that 0 K corresponds to -273.15 °C and that a 1 K change equals a 1 °C change. The relationship can be expressed as: \( T(K) = T(°C) + 273.15 \).

Step 2: Convert Nitrogen Boiling Point to Celsius

Given the boiling point of nitrogen is 77.4 K, use the conversion equation: \( T(°C) = T(K) - 273.15 \). For nitrogen, \( T(°C) = 77.4 - 273.15 = -195.75 \) °C.

Step 3: Convert Nitrogen Boiling Point to Fahrenheit

To convert from Celsius to Fahrenheit, use the equation \( T(°F) = \frac{9}{5}T(°C) + 32 \). For nitrogen, \( T(°F) = \frac{9}{5} \times (-195.75) + 32 = -320.35 \) °F.

Step 4: Convert Oxygen Boiling Point to Celsius

Given the boiling point of oxygen is 90.2 K, use the conversion equation: \( T(°C) = 90.2 - 273.15 = -182.95 \) °C.

Step 5: Convert Oxygen Boiling Point to Fahrenheit

Convert from Celsius to Fahrenheit using the equation: \( T(°F) = \frac{9}{5}T(°C) + 32 \). For oxygen, \( T(°F) = \frac{9}{5} \times (-182.95) + 32 = -297.31 \) °F.

Step 6: Determine Which Element Distills First

Nitrogen has a lower boiling point than oxygen, both in Celsius and Fahrenheit. Therefore, nitrogen will distill first as its boiling point is reached earlier when the temperature of the liquid mixture is increased.

Key Concepts

Understanding Thermodynamics: The Study of Energy

Thermodynamics is the branch of physics that deals with heat, work, and the forms of energy involved in physical and chemical processes. It's all about energy change. Let's break down the core aspects of thermodynamics that relate to temperature changes and boiling points.

• Energy Transfer: Thermodynamics describes how heat energy is transferred between systems. For example, when you heat a liquid, its molecules move faster, leading to increased temperature and changes of state, such as from liquid to gas at the boiling point.
• States of Matter: Understanding how substances change from solid to liquid to gas (and vice-versa) is vital in thermodynamics. The amount of heat energy needed for these transformations is a key focus.

The exercise demonstrates this by pointing out how liquefied air is warmed to separate nitrogen and oxygen based on their boiling points. When each gas reaches its specific boiling point, it changes from liquid to gas, illustrating practical applications of thermodynamic principles.

Temperature Scales: Measuring Heat Precisely

Temperature scales allow us to measure and understand how hot or cold something is. The three most commonly used scales are Celsius, Fahrenheit, and Kelvin. Each scale has unique characteristics that make it suitable for different applications.

• Celsius: Widely used around the world, particularly in scientific contexts, the Celsius scale sets the freezing point of water at 0°C and the boiling point at 100°C.
• Fahrenheit: Predominantly used in the United States, the Fahrenheit scale places the freezing point of water at 32°F and the boiling point at 212°F, providing a greater range of degrees for everyday temperature differences.
• Kelvin: An absolute temperature scale, Kelvin is used in scientific calculations. Zero Kelvin (0 K) is absolute zero, where all molecular motion stops. The Kelvin scale is straightforward to convert from Celsius, as it only requires adding 273.15 to the Celsius temperature.

In exploring the solution, you apply the Kelvin-to-Celsius conversion formula to translate boiling points. This method simplifies understanding essential transformations between temperature scales.

Boiling Point: The Gateway to State Changes

The boiling point of a substance is the temperature at which it changes from a liquid to a gas. This concept is pivotal in processes such as distillation.

Why Boiling Points Matter

Boiling points can be affected by pressure. For instance, water boils at lower temperatures at higher altitudes where the atmospheric pressure is reduced. In industrial processes, controlling boiling points allows for the separation and purification of chemical compounds.

• Practical Example: In the specified problem, the boiling points of nitrogen and oxygen are exploited. Nitrogen boils at 77.4 K, while oxygen boils at 90.2 K, allowing them to be separated through controlled warming of liquefied air.
• Differentiation of Elements: Since nitrogen has a lower boiling point than oxygen, it transitions to a gaseous state first when heated, highlighting its distillation ahead of oxygen.

By comparing these boiling points, the exercise illustrates how boiling points play a critical role in practical thermodynamic applications.