A blown up balloon has a volume of 500 mL at 278K. The balloon is distended to 7/8th of it's max stretching capacity. Will the balloon burst at 303K. Determine the minimum temp above which it'll burst?

To determine whether the balloon will burst at 303K and to find the minimum temperature above which it will burst, we can use the ideal gas law and the concept of volume expansion. The ideal gas law states that for a given amount of gas at constant pressure, the volume of the gas is directly proportional to its temperature in Kelvin. This relationship can be expressed as:

Understanding the Problem

We start with a balloon that has a volume of 500 mL at a temperature of 278K. When the balloon is distended to 7/8 of its maximum capacity, we need to calculate the new volume and then see how the temperature affects the pressure inside the balloon.

Step 1: Calculate the Maximum Volume

Let’s denote the maximum volume of the balloon as V_max. Since the balloon is currently at 7/8 of its maximum capacity, we can express this as:

• Current Volume (V) = (7/8) * V_max

Given that the current volume (V) is 500 mL, we can rearrange the equation to find V_max:

• 500 mL = (7/8) * V_max
• V_max = 500 mL * (8/7) = approximately 571.43 mL

Step 2: Determine the Volume at 303K

Using the ideal gas law, we can relate the volumes and temperatures. The relationship can be expressed as:

• (V1 / T1) = (V2 / T2)

Where:

• V1 = 500 mL (initial volume)
• T1 = 278 K (initial temperature)
• V2 = ? (volume at 303K)
• T2 = 303 K (final temperature)

Rearranging the equation to solve for V2 gives us:

• V2 = V1 * (T2 / T1)
• V2 = 500 mL * (303 K / 278 K) = approximately 544.25 mL

Step 3: Compare with Maximum Volume

Now we need to compare this volume (544.25 mL) with the maximum volume (571.43 mL). Since 544.25 mL is less than 571.43 mL, the balloon will not burst at 303K.

Finding the Minimum Temperature for Bursting

To find the minimum temperature above which the balloon will burst, we need to set the volume equal to the maximum volume:

• V_max = V1 * (T_burst / T1)

Rearranging gives us:

• T_burst = V_max * (T1 / V1)

Substituting the known values:

• T_burst = 571.43 mL * (278 K / 500 mL)
• T_burst = 571.43 * 0.556 = approximately 318.18 K

Final Thoughts

Thus, the balloon will burst at temperatures above approximately 318.18 K. Therefore, at 303K, the balloon is safe, but if the temperature rises above 318.18 K, it risks bursting due to the increased pressure from the gas expanding within the balloon.