Can you help me please..
S'mores are an American treat that consists of a roasted marshmallow and some chocolate sandwiched between two graham crackers. Ideally, the chocolate is not so solid that it is difficult to bite through but also solid enough that it doesn't melt all over your hand when you take a bite out of your s'more.
Chocolate begins to melt between 40°C and 45°C depending on the different components of chocolate (which we will study in a later week). Assume that the melting temperature for chocolate is 43°C, and that heat transfer between the marshmallow and chocolate is perfectly efficient. Given this, to what temperature should you cool the roasted marshmallow (7.5 g) before adding it to the s'more so that the chocolate (50 g) is warmed up to 43°C from 25°C? Use 2 J/g⋅°C as the specific heat for both the chocolate and the marshmallow.
Enter the ideal temperature of the marshmallow in degrees Celsius (but without including the units):
50 * (43 - 25) = 7.5 * (t - 43) ... solve for t
the specific heats are the same, so they cancel
solving the calometry
To find the ideal temperature of the marshmallow, we need to calculate the amount of heat transferred from the marshmallow to the chocolate.
First, let's calculate the heat transfer for the chocolate. We know the specific heat capacity of the chocolate is 2 J/g⋅°C, the mass of the chocolate is 50 g, and we want to warm it up from 25°C to 43°C. The formula to calculate the heat transfer is:
Q = m * c * ΔT
where Q is the heat transfer, m is the mass, c is the specific heat capacity, and ΔT is the change in temperature.
For the chocolate:
Q_chocolate = 50 g * 2 J/g⋅°C * (43°C - 25°C)
Q_chocolate = 50 g * 2 J/g⋅°C * 18°C
Q_chocolate = 1800 J
Since the heat transfer from the marshmallow to the chocolate is perfectly efficient, the heat transferred from the marshmallow is equal to the heat transferred to the chocolate. Therefore, the heat transferred from the marshmallow is also 1800 J.
Now, let's calculate the heat transfer for the marshmallow. We know the specific heat capacity of the marshmallow is also 2 J/g⋅°C, and the mass of the marshmallow is 7.5 g. We want to find the change in temperature for the marshmallow, so rearranging the formula:
Q_marshmallow = m * c * ΔT
ΔT = Q_marshmallow / (m * c)
ΔT = 1800 J / (7.5 g * 2 J/g⋅°C)
ΔT = 20°C
Therefore, you should cool the roasted marshmallow to 20°C before adding it to the s'more so that the chocolate is warmed up to 43°C.
To find the ideal temperature of the marshmallow, we need to consider the heat transfer between the marshmallow and the chocolate.
First, let's calculate the heat transferred when the marshmallow is cooled down using the formula:
Q = mcΔT
Where:
Q is the heat transferred
m is the mass of the marshmallow (7.5 g)
c is the specific heat of the marshmallow (2 J/g⋅°C)
ΔT is the change in temperature
We want to find ΔT, which is the change in temperature to cool the marshmallow down to the ideal temperature. We'll assume it starts at room temperature (25°C) and is cooled down to the ideal temperature.
Next, let's calculate the heat transferred when the chocolate is warmed up using the same formula:
Q = mcΔT
Where:
Q is the heat transferred
m is the mass of the chocolate (50 g)
c is the specific heat of the chocolate (2 J/g⋅°C)
ΔT is the change in temperature (from 25°C to 43°C)
We can equate the two equations since the heat transferred from the marshmallow to the chocolate is the same:
m_marshmallow * c_marshmallow * ΔT_marshmallow = m_chocolate * c_chocolate * ΔT_chocolate
Plugging in the given values:
(7.5 g) * (2 J/g⋅°C) * ΔT_marshmallow = (50 g) * (2 J/g⋅°C) * (43°C - 25°C)
Simplifying:
(7.5 g) * (2 J/g⋅°C) * ΔT_marshmallow = (50 g) * (2 J/g⋅°C) * (18°C)
Canceling out the units:
(7.5) * ΔT_marshmallow = (50) * (18)
Solving for ΔT_marshmallow:
ΔT_marshmallow = (50) * (18) / (7.5)
ΔT_marshmallow = 120°C
Finally, to find the ideal temperature of the marshmallow, we subtract the change in temperature from the starting temperature:
Ideal temperature of the marshmallow = 25°C - 120°C
Ideal temperature of the marshmallow = -95°C
However, it's important to note that the calculated ideal temperature is below freezing, which is not practical or safe for consumption. The equation assumes perfect efficiency and does not take into account practical considerations such as the limitations of heat transfer or food safety. Therefore, it is not possible to achieve the desired temperature based on the given information.