Problem 2 - College 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 (d) 277.15 K, 39.2°F; 113.15 K, -256°F (e) 40.56°C, 313.71 K.

Step 1: Convert from Kelvin to Celsius and Fahrenheit (Part a)

To convert from Kelvin to Celsius, use the formula: \[T_{C} = T_{K} - 273.15\]For Kelvin to Fahrenheit, use:\[T_{F} = \frac{9}{5}(T_{K} - 273.15) + 32\]Applying the formulas for \( T_{K} = 310 \) K:\[T_{C} = 310 - 273.15 = 36.85^{\circ} \text{C}\]\[T_{F} = \frac{9}{5}(36.85) + 32 = 98.33^{\circ} \text{F}\]

Step 2: Convert from Celsius to Kelvin and Fahrenheit (Part b)

To convert from Celsius to Kelvin, use the formula:\[T_{K} = T_{C} + 273.15\]For Celsius to Fahrenheit, use:\[T_{F} = \frac{9}{5}T_{C} + 32\]Applying these to \( T_{C} = 40 \) \,\ ^{\circ}\text{C}:\[T_{K} = 40 + 273.15 = 313.15 \, \text{K}\]\[T_{F} = \frac{9}{5}(40) + 32 = 104 \, ^{\circ}\text{F}\]

Step 3: Convert Temperature Difference to Kelvin and Fahrenheit (Part c)

The temperature difference in Celsius is 7 \, ^{\circ}\text{C}. To convert a temperature difference from Celsius to Kelvin, note that the magnitude is the same: 7 \, \text{K}.For Fahrenheit difference, use:\[\Delta T_{F} = \frac{9}{5} \times \Delta T_{C}\]So,\[\Delta T_{F} = \frac{9}{5} \times 7 = 12.6 \, ^{\circ}\text{F}\]

Step 4: Convert Celsius to Kelvin and Fahrenheit (Part d - Blood Storage)

For both temperature conversions:- \(4.0^{\circ} \text{C} \): - \(T_{K} = 4.0 + 273.15 = 277.15 \) \, \text{K} - \(T_{F} = \frac{9}{5}(4.0) + 32 = 39.2 \,^{\circ}\text{F}\)- \(-160^{\circ} \text{C} \): - \(T_{K} = -160 + 273.15 = 113.15 \) \, \text{K} - \(T_{F} = \frac{9}{5}(-160) + 32 = -256 \,^{\circ}\text{F}\)

Step 5: Convert Fahrenheit to Celsius and Kelvin (Part e)

To convert 105^{\circ} \text{F} to Celsius:\[T_{C} = \frac{5}{9}(T_{F} - 32)\]For Kelvin, use:\[T_{K} = \frac{5}{9}(T_{F} - 32) + 273.15\]Using \(T_{F} = 105^{\circ} \text{F}\):\[T_{C} = \frac{5}{9}(105 - 32) = 40.56^{\circ} \text{C}\]\[T_{K} = 40.56 + 273.15 = 313.71\ \, \text{K}\]

Key Concepts

Biomedicine Temperature

In the realm of biomedicine, temperature plays a crucial role in understanding the state and conditions of the human body. Temperatures can indicate a state of health or illness, and knowing how to convert these temperatures between different scales can be pivotal. For example, medical thermometers measure body temperature in either Celsius, Kelvin, or Fahrenheit, depending on the context. This is important for diagnosing conditions like fever, hypothermia, or hyperthermia. Awareness of normal and critical temperature ranges can aid in quick diagnosis and treatment.

Celsius to Kelvin

Conversion between Celsius and Kelvin is straightforward and essential in both clinical and research settings. The Kelvin scale is commonly used in scientific measurements because it starts at absolute zero, making it a well-suited scale for many scientific calculations. The conversion formula from Celsius to Kelvin is \[ T_{K} = T_{C} + 273.15 \]This equation indicates that to convert from Celsius to Kelvin, you add 273.15 to the Celsius temperature. This simple addition is used frequently in calculations where absolute temperature is necessary, such as studies on thermodynamics and heat transfer in biological systems.

Fahrenheit Conversion

Converting temperatures to and from Fahrenheit is particularly common in the medical field in regions like the United States. The formula to convert Celsius to Fahrenheit is: \[ T_{F} = \frac{9}{5}T_{C} + 32 \]This formula helps convert the Celsius reading often used in scientific discussions to Fahrenheit, which might be used at a local clinical setting. The reverse can be done using: \[ T_{C} = \frac{5}{9}(T_{F} - 32) \]By understanding these formulas, medical professionals and researchers can easily interpret temperature readings, which is crucial for providing effective care and conducting accurate scientific studies.

Heat Stroke Temperature

Heat stroke is a serious condition that occurs when the body overheats, typically above 105°F (or 40.56°C and 313.71 K). It is vital to recognize these temperature thresholds to prevent potential damage to the brain or other vital organs. Higher body temperatures during hot weather or intense exercise require immediate attention. Recognizing the symptoms and understanding temperature readings using various scales can help initiate prompt treatment.

• Immediate cooling methods, such as ice packs or cold baths, can lower body temperature.
• Seeking medical attention quickly can prevent lasting health effects.

Accurate and fast conversions are influential during emergencies, helping in understanding the severity and aiding in consistent communication in healthcare settings.