Problem 2 - University Physics with Modern Physics

Question

BIO Temperatures in Biomedicine. (a) Normal body temperature. The average normal body temperature measured in the mouth is 310 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\(^\circ\)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 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\(^\circ\)C lasts safely for about 3 weeks, whereas blood stored at -160\(^\circ\)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\(^\circ\)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; (d) 277.15 K, 39.2°F for 4°C; 113.15 K, -256°F for -160°C; (e) 40.56°C, 313.71 K.

Step 1: Convert Kelvin to Celsius

To convert from Kelvin to Celsius, use the formula: \[ T(^{\circ}C) = T(K) - 273.15 \] Given \( T = 310 \ K \), the conversion is: \( T = 310 - 273.15 = 36.85^{\circ}C \).

Step 2: Convert Celsius to Fahrenheit

The formula to convert Celsius to Fahrenheit is: \[ T(^{\circ}F) = T(^{\circ}C) \times \frac{9}{5} + 32 \]Using \( T = 36.85^{\circ}C \), the conversion becomes:\[ T = 36.85 \times \frac{9}{5} + 32 \approx 98.33^{\circ}F \].

Step 3: Convert Celsius to Kelvin for (b)

For \( 40^{\circ}C \), using the formula \[ T(K) = T(^{\circ}C) + 273.15 \],we get \( T = 40 + 273.15 = 313.15 \ K \).

Step 4: Convert Celsius to Fahrenheit for (b)

Using the temperature \( T = 40^{\circ}C \) and the formula: \[ T(^{\circ}F) = 40 \times \frac{9}{5} + 32 \],we find \( T = 104^{\circ}F \).

Step 5: Temperature Difference in Kelvin

A temperature difference of 7 degrees Celsius can be expressed in Kelvin as the same numerical value, since both increments are equal. So, it is \( 7 \ K \).

Step 6: Temperature Difference in Fahrenheit

For temperature differences, use \[ \Delta T(^{\circ}F) = \Delta T(^{\circ}C) \times \frac{9}{5} \],so \( \Delta T = 7 \times \frac{9}{5} = 12.6^{\circ}F \).

Step 7: Convert Celsius to Kelvin for Blood Storage

Convert \(4^{\circ}C\) to Kelvin: \[ T(K) = 4 + 273.15 = 277.15 \ K \].Convert \(-160^{\circ}C\) to Kelvin: \[ T(K) = -160 + 273.15 = 113.15 \ K \].

Step 8: Convert Celsius to Fahrenheit for Blood Storage

Convert \(4^{\circ}C\) to Fahrenheit: \[ T(^{\circ}F) = 4 \times \frac{9}{5} + 32 = 39.2^{\circ}F \].Convert \(-160^{\circ}C\) to Fahrenheit: \[ T(^{\circ}F) = -160 \times \frac{9}{5} + 32 = -256^{\circ}F \].

Step 9: Convert Fahrenheit to Celsius for Heat Stroke

Using \( 105^{\circ}F \) and the formula: \[ T(^{\circ}C) = \frac{105 - 32}{1.8} \approx 40.56^{\circ}C \].

Step 10: Convert Celsius to Kelvin for Heat Stroke

For \( 40.56^{\circ}C \): \[ T(K) = 40.56 + 273.15 \approx 313.71 \ K \].

Key Concepts

Celsius to Fahrenheit conversion

Converting Celsius to Fahrenheit is an essential skill in many practical situations. This conversion helps us in understanding temperature measurements in regions where the Fahrenheit scale is predominantly used. The formula to convert a temperature in Celsius (°C) to Fahrenheit (°F) is:

• \( T(^{\circ}F) = T(^{\circ}C) \times \frac{9}{5} + 32 \)

For instance, if we take a temperature of 36.85°C, which is a typical human body temperature, the Fahrenheit equivalent would be calculated as follows:

• \( T(^{\circ}F) = 36.85 \times \frac{9}{5} + 32 \approx 98.33^{\circ}F \)

This demonstrates that 36.85°C is approximately 98.33°F, a familiar benchmark for a normal body temperature in the context of healthcare in the United States.
Understanding these conversions is crucial in medical fields, as patient records and prescriptions often require temperature readings in a specific scale.

Celsius to Kelvin conversion

Celsius and Kelvin are both metric scales often used in scientific contexts. The Kelvin scale is especially useful in laboratory settings, where absolute temperature is required. To convert from Celsius to Kelvin, the formula is straightforward:

• \( T(K) = T(^{\circ}C) + 273.15 \)

For example, if a body's surface temperature is 40°C, converting this to Kelvin involves:

• \( T(K) = 40 + 273.15 = 313.15 \ K \)

This conversion shows the temperature in Kelvin, which is often used in research and core scientific studies as it starts at absolute zero, where all molecular motion stops.
In biomedicine and physics, accurate temperature measurements in Kelvin are critical for experiments that require exact and consistent baseline measurements.

temperature differences

Temperature differences can be quantified in various temperature scales. In bio-medicine, understanding these differences is useful for interpreting clinical data. For example, if an internal body temperature is 7°C higher than the surface temperature, this difference can be expressed in Kelvin as:

• \( \Delta T = 7 \ K \)

Kelvin and Celsius increments are identical, so the difference is numerically the same. To express this difference in Fahrenheit, we use another conversion formula:

• \( \Delta T(^{\circ}F) = \Delta T(^{\circ}C) \times \frac{9}{5} \)
• \( \Delta T = 7 \times \frac{9}{5} = 12.6^{\circ}F \)

Understanding temperature differences in various scales is important in diagnosing and treating conditions that involve heat regulation in the body.

applications in biomedicine

Temperature is a critical parameter in biomedicine, affecting everything from patient diagnosis to biomedical storage systems. In situations such as blood storage, precise temperature control can drastically affect the safety and longevity of biological samples. Blood, for instance, needs to be stored at very specific temperatures. At 4°C, blood is safe for about 3 weeks. This temperature translates to 277.15 K or approximately 39.2°F.
Conversely, at ultra-cold conditions of -160°C, equivalent to 113.15 K or -256°F, blood can last significantly longer, up to 5 years. Such precision in temperature measurement and control is vital in bio-laboratories and hospitals.
Even in clinical scenarios like monitoring for heat stroke, understanding exact temperature conversions can be lifesaving. The threshold of 105°F related to heat stroke risks translates to approximately 40.56°C or 313.71 K.
These conversions are crucial to ensure that medical professionals can interpret and apply temperature data across different systems, aiding diagnostics and enhancing patient care.