Hi Kevin!
Great question! We can solve this using Charles's Law, which tells us that when the pressure of a gas stays constant, its volume increases as its temperature increases.
Let's work through it step by step!
Step 1: Identify the information given
We know:
- Initial volume ((V_1)) = 200 mL
- Initial temperature ((T_1)) = Unknown
- Final volume ((V_2)) = 120 mL
- Final temperature ((T_2)) = 33°C
We're solving for the initial temperature.
Step 2: Apply Charles's Law
Charles's Law states:
[
\frac{V_1}{T_1}=\frac{V_2}{T_2}
]
Important: Temperatures must be expressed in Kelvin, not Celsius!
Step 3: Convert Celsius to Kelvin
Use:
[
K=°C+273.15
]
Therefore:
[
T_2=33+273.15
]
[
\boxed{T_2=306.15\text{ K}}
]
Step 4: Rearrange the formula to solve for the initial temperature
Starting with:
[
\frac{V_1}{T_1}=\frac{V_2}{T_2}
]
Cross-multiply:
[
V_1T_2=V_2T_1
]
Divide both sides by (V_2):
[
T_1=\frac{V_1T_2}{V_2}
]
Now substitute:
[
T_1=\frac{200(306.15)}{120}
]
[
T_1=\frac{61,230}{120}
]
[
\boxed{T_1=510.25\text{ K}}
]
Step 5: Convert Kelvin back to Celsius
Since the question asks for the temperature in degrees Celsius:
[
°C=K-273.15
]
[
T_1=510.25-273.15
]
[
\boxed{T_1=237.10°C}
]
Final Answer
[
\boxed{237.1°C}
]
Does Our Answer Make Sense?
Absolutely!
The balloon originally had a volume of 200 mL.
After cooling, its volume decreased to 120 mL.
According to Charles's Law, when temperature decreases, gas volume also decreases, provided pressure remains constant.
Therefore, the initial temperature must have been higher than the final temperature of 33°C.
Our answer of approximately 237°C makes sense!
Helpful tip: Whenever you use Charles's Law, always convert temperatures to Kelvin before substituting into the formula. This is one of the most common mistakes students make with gas-law problems.
Hope this helps!
Michael
