Problem 2 - University Physics with Modern Physics

Question

Temperatures in Biomedicine. (a) Normal body temperature. The average normal body temperature measured in the mouth is 310 \(\mathrm{K}\) . What would Celsius and Fahrenheit thermometers read for this temperature? (b) Elevated body temperature. During very vigorous exercise, the body's temperature can go as high as \(40^{\circ} \mathrm{C} .\) What would Kelvin and Fahrenheit thermometers read for this temperature? (c) Temperature difference in the body. The surface temperature of the body is normally about 7 \(\mathrm{C}^{\circ}\) lower than the internal temperature. Express this temperature difference in kelvins and in Fahrenheit degrees. (d) Blood storage. Blood stored at \(4.0^{\circ} \mathrm{C}\) lasts safely for about 3 weeks, whereas blood stored at \(-160^{\circ} \mathrm{C}\) lasts for 5 years. Express both temperatures on the Fahrenheit and Kelvin scales. (e) Heat stroke. If the body's temperature is above \(105^{\circ} \mathrm{F}\) for a prolonged period, heat stroke can result. Express this temperature on the Celsius and Kelvin scales.

Step-by-Step Solution

Answer

(a) 36.85°C, 98.33°F; (b) 313.15 K, 104°F; (c) 7 K, 12.6°F difference; (d) 39.2°F, 277.15 K and -256°F, 113.15 K; (e) 40.56°C, 313.71 K.

Step 1: Convert 310 K to Celsius

To convert temperature from Kelvin to Celsius, use the formula: \[ T_{\text{C}} = T_{\text{K}} - 273.15 \]Substituting the given temperature in Kelvin:\[ T_{\text{C}} = 310 - 273.15 = 36.85^{\circ} \text{C} \]

Step 2: Convert 310 K to Fahrenheit

To convert Celsius to Fahrenheit, the formula is:\[ T_{\text{F}} = T_{\text{C}} \times \frac{9}{5} + 32 \]Using the Celsius result from Step 1:\[ T_{\text{F}} = 36.85 \times \frac{9}{5} + 32 = 98.33^{\circ} \text{F} \]

Step 3: Convert 40°C to Kelvin

To convert Celsius to Kelvin, add 273.15:\[ T_{\text{K}} = 40 + 273.15 = 313.15\,\text{K} \]

Step 4: Convert 40°C to Fahrenheit

Using the conversion formula:\[ T_{\text{F}} = 40 \times \frac{9}{5} + 32 = 104^{\circ} \text{F} \]

Step 5: Convert 7°C difference to Kelvin and Fahrenheit

A difference in temperature is the same numerical change in Kelvin as in Celsius, since:\[ \Delta T_{\text{K}} = \Delta T_{\text{C}} = 7 \]\Convert to Fahrenheit:\[ \Delta T_{\text{F}} = \Delta T_{\text{C}} \times \frac{9}{5} = 7 \times \frac{9}{5} = 12.6^{\circ} \text{F} \]

Step 6: Convert 4°C to Fahrenheit and Kelvin

First, convert to Fahrenheit:\[ T_{\text{F}} = 4 \times \frac{9}{5} + 32 = 39.2^{\circ} \text{F} \]Convert to Kelvin:\[ T_{\text{K}} = 4 + 273.15 = 277.15\,\text{K} \]

Step 7: Convert -160°C to Fahrenheit and Kelvin

First, convert to Fahrenheit:\[ T_{\text{F}} = (-160) \times \frac{9}{5} + 32 = -256^{\circ} \text{F} \]Convert to Kelvin:\[ T_{\text{K}} = -160 + 273.15 = 113.15\,\text{K} \]

Step 8: Convert 105°F to Celsius and Kelvin

Convert Fahrenheit to Celsius:\[ T_{\text{C}} = \frac{105 - 32}{1.8} = 40.56^{\circ} \text{C} \] Convert Celsius to Kelvin:\[ T_{\text{K}} = 40.56 + 273.15 = 313.71\,\text{K} \]

Key Concepts

Kelvin

Kelvin is the SI unit for temperature, often used in scientific contexts. Unlike Celsius or Fahrenheit, Kelvin doesn't use degrees. It's an absolute scale, meaning it starts from absolute zero, the theoretically coldest possible temperature. To convert a temperature from Kelvin to Celsius, you subtract 273.15, as Kelvin and Celsius scales share the same degree size and zero point. This direct relation makes Kelvin perfect for scientific calculations.
For instance:

• To convert 310 K to Celsius: \( T_C = 310 - 273.15 = 36.85^{\circ} C \)
• To convert back to Kelvin from Celsius: add 273.15

The Kelvin scale is widely used in various scientific fields, including physics and chemistry, due to its use of absolute zero.

Celsius

The Celsius scale is a temperature scale named after inventor Anders Celsius. It is commonly used worldwide for most temperature measurements. In this scale, water freezes at 0°C and boils at 100°C under standard atmospheric conditions. This centigrade scale is intuitive because it is based on the properties of water, making it practical for daily use.
For conversions:

• From Celsius to Kelvin: Add 273.15.
• From Celsius to Fahrenheit: Multiply by \( \frac{9}{5} \), then add 32.

For example, if a temperature is 40°C, to convert to Fahrenheit, you would multiply by 1.8 (or 9/5) and add 32, resulting in 104°F. The simplicity of this scale makes it a favorite for everyday temperature settings. Health fields, like medicine, often use Celsius for body temperature readings.

Fahrenheit

Fahrenheit is a temperature scale named after Daniel Gabriel Fahrenheit. It is primarily used in the United States. In this scale, water freezes at 32°F and boils at 212°F, a range divided into 180 equal divisions, called degrees. This contrasts with the 100 divisions in Celsius.
To convert Celsius to Fahrenheit, use:

• \( T_{F} = T_{C} \times \frac{9}{5} + 32 \)

For example, a temperature of 36.85°C, derived from 310 K, converts to approximately 98.33°F, close to the average human body temperature. Many use Fahrenheit for everyday weather forecasts in the USA due to its traditional usage and cultural precedent.

Biomedicine Temperatures

Temperature conversion is crucial in biomedicine, where precision is necessary to ensure correct diagnosis and treatment. Normal human body temperature ranges around 36.85°C or 98.33°F, while extreme temperatures can signal health issues.
In biomedicine:

• At elevated body temperatures, like 40°C, conversion to other scales confirms an increase: 104°F in Fahrenheit, 313.15 K in Kelvin.
• Such precision is vital when monitoring conditions like hypothermia and hyperthermia, which require conversion for global comprehension.

Temperature conversion in medicine helps standardize readings. It prevents potential miscommunication between professionals in different parts of the world using various scales. Blood storage and understanding critical temperatures (e.g., heat stroke at 105°F) also leverage these conversions. Ensuring compatibility across Celsius, Kelvin, and Fahrenheit empowers healthcare providers to maintain accurate and safe practices.