📘 Solved Numerical Problems (Thermal Properties of Matter)
Related Resources:
Important Formulas
Celsius to Fahrenheit Conversion
or
\[ T_F = 1.8 T_c + 32 \]Celsius to Kelvin Conversion
Thermometer Linear Scale Formula
Where \( l_0 \) is the length of mercury thread and \( l_{100} \) is the length between fixed points
Solved Numerical Problems
Given:
Normal human body temperature is 98.6°F. Convert it into Celsius scale and Kelvin scale.
\[ \text{Temperature in Fahrenheit} = T_F = 98.6°F \]
Solution:
Using Fahrenheit to Celsius conversion:
\[ T_F = 1.8 T_c + 32 \]
\[ 98.6 = 1.8 T_c + 32 \]
\[ 98.6 - 32 = 1.8 T_c \]
\[ 66.6 = 1.8 T_c \]
\[ T_c = \frac{66.6}{1.8} \]
\[ T_c = 37°C \]
Now using Celsius to Kelvin conversion:
\[ T_K = 273 + T_c \]
\[ T_K = 273 + 37 \]
\[ T_K = 310 K \]
Answer: Normal human body temperature is 37°C and 310 K.
Given:
Find the temperature at which Celsius and Fahrenheit thermometer readings would be the same.
Let temperature in Celsius = \( T_c = T \)
And temperature in Fahrenheit = \( T_F = T \)
Solution:
Using the conversion formula:
\[ T_F = \frac{9}{5} T_c + 32 \]
\[ T = \frac{9}{5} T + 32 \]
\[ T - \frac{9}{5}T = 32 \]
\[ \frac{5T - 9T}{5} = 32 \]
\[ -\frac{4T}{5} = 32 \]
\[ -4T = 160 \]
\[ T = -40 \]
Answer: Celsius and Fahrenheit scales show the same reading at -40°.
Given:
Convert 5°F to Celsius and Kelvin scales.
\[ \text{Temperature in Fahrenheit} = T_F = 5°F \]
Solution:
Using Fahrenheit to Celsius conversion:
\[ T_F = 1.8 T_c + 32 \]
\[ 5 = 1.8 T_c + 32 \]
\[ 5 - 32 = 1.8 T_c \]
\[ -27 = 1.8 T_c \]
\[ T_c = \frac{-27}{1.8} \]
\[ T_c = -15°C \]
Now using Celsius to Kelvin conversion:
\[ T_K = 273 + T_c \]
\[ T_K = 273 + (-15) \]
\[ T_K = 273 - 15 \]
\[ T_K = 258 K \]
Answer: 5°F is equal to -15°C and 258 K.
Given:
Find the equivalent temperature of 25°C on Fahrenheit and Kelvin scales.
\[ \text{Temperature in Celsius} = T_c = 25°C \]
Solution:
Using Celsius to Fahrenheit conversion:
\[ T_F = 1.8 T_c + 32 \]
\[ T_F = 1.8 \times 25 + 32 \]
\[ T_F = 45 + 32 \]
\[ T_F = 77°F \]
Using Celsius to Kelvin conversion:
\[ T_K = 273 + T_c \]
\[ T_K = 273 + 25 \]
\[ T_K = 298 K \]
Answer: 25°C is equal to 77°F and 298 K.
Given:
The ice and steam points on an ungraduated thermometer are found to be 192 mm apart. What temperature will be on Celsius scale if the length of mercury thread is at 67.2 mm above the ice point mark?
\[ \text{Length between ice and steam points} = l_{100} = 192 \text{ mm} \]
\[ \text{Length of mercury thread} = l_0 = 67.2 \text{ mm} \]
Solution:
Using the linear scale formula for thermometers:
\[ T_c = \frac{l_0}{l_{100}} \times 100 \]
\[ T_c = \frac{67.2}{192} \times 100 \]
\[ T_c = 0.35 \times 100 \]
\[ T_c = 35°C \]
Answer: The temperature on the Celsius scale is 35°C.
Given:
The length between the fixed points of liquid-in-glass thermometer is 20 cm. If the mercury level is 4.5 cm above the lower mark, what is the temperature on the Fahrenheit scale?
\[ \text{Length between fixed points} = l_{100} = 20 \text{ cm} \]
\[ \text{Length of mercury thread} = l_0 = 4.5 \text{ cm} \]
Solution:
First, find the temperature in Celsius:
\[ T_c = \frac{l_0}{l_{100}} \times 100 \]
\[ T_c = \frac{4.5}{20} \times 100 \]
\[ T_c = 0.225 \times 100 \]
\[ T_c = 22.5°C \]
Now convert to Fahrenheit:
\[ T_F = 1.8 T_c + 32 \]
\[ T_F = 1.8 \times 22.5 + 32 \]
\[ T_F = 40.5 + 32 \]
\[ T_F = 72.5°F \]
Answer: The temperature on the Fahrenheit scale is 72.5°F.
Practice Problems
Try solving these problems on your own:
- Convert 100°C to Fahrenheit and Kelvin scales.
- Convert 0 K to Celsius and Fahrenheit scales.
- The length between fixed points of a thermometer is 25 cm. If the mercury thread is 18 cm above the lower fixed point, what is the temperature in Celsius and Fahrenheit?
- At what temperature will the Fahrenheit reading be double the Celsius reading?
- A thermometer has its ice point marked as 10° and steam point as 90°. If this thermometer reads 50°, what is the actual temperature in Celsius?
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