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Step-by-Step Solution
Step 1: Convert given temperatures to Kelvin
The initial temperature is given as 17°C and the final temperature is 27°C. To convert from °C to K, we add 273:
Initial temperature,
$ T_i = 17 + 273 = 290 \, \text{K} $
Final temperature,
$ T_f = 27 + 273 = 300 \, \text{K} $
Step 2: Recall the ideal gas relation for number of molecules
From the ideal gas equation in terms of number of molecules N,
$ PV = \frac{N}{N_A} \, R \, T, $
where
$N$ is the number of molecules,
$N_A$ is Avogadro’s number,
$R$ is the universal gas constant,
$P$ is the pressure,
$V$ is the volume,
and
$T$ is the temperature in Kelvin.
Rearranging for $N$:
$ N = \frac{P \, V \, N_A}{R \, T}. $
Step 3: Express the difference in number of molecules, $N_f - N_i$
Before heating, the number of molecules in the room is
$ N_i = \frac{P_0 \, V_0 \, N_A}{R \, T_i}. $
After heating, the number of molecules is
$ N_f = \frac{P_0 \, V_0 \, N_A}{R \, T_f}. $
Hence, the difference $ \bigl(N_f - N_i\bigr) $ is
$ N_f - N_i
= \frac{P_0 \, V_0 \, N_A}{R} \biggl(\frac{1}{T_f} - \frac{1}{T_i}\biggr). $
Step 4: Substitute the known values and compute
Atmospheric pressure,
$ P_0 = 1 \times 10^5 \,\text{Pa}$
Volume of the room,
$ V_0 = 30 \,\text{m}^3$
Avogadro’s number,
$ N_A = 6.023 \times 10^{23} \,\text{mol}^{-1}$
Universal gas constant,
$ R = 8.314 \,\text{J} \,\text{mol}^{-1}\,\text{K}^{-1}$
$ T_i = 290 \,\text{K}, \quad T_f = 300 \,\text{K}$
Thus,
$ N_f - N_i
= \frac{(1 \times 10^5) \,\,(30)\,\,(6.023 \times 10^{23})}{8.314}
\biggl(\frac{1}{300} - \frac{1}{290}\biggr). $
First, compute the bracket:
$ \left(\frac{1}{300} - \frac{1}{290}\right)
= \frac{290 - 300}{290 \times 300}
= \frac{-10}{87000}
= -1.149 \times 10^{-4} \approx. $
Then multiply by the prefactor:
$ \frac{1 \times 10^5 \times 30 \times 6.023 \times 10^{23}}{8.314}
\approx 2.17 \times 10^{29}. $
Finally,
$ N_f - N_i
\approx 2.17 \times 10^{29} \times \bigl(-1.149 \times 10^{-4}\bigr)
= -2.5 \times 10^{25}. $
Step 5: Interpret the result
The negative sign indicates that for the pressure and volume to remain constant while temperature increases, the net number of molecules inside the room decreases. Therefore,
$ N_f - N_i \approx -2.5 \times 10^{25}. $
Although the question statement lists “2.5 × 1025” as the correct answer, the calculation from the ideal gas relation shows a negative sign, meaning there is a reduction in the number of molecules. The magnitude of change is
$2.5 \times 10^{25}$.
