Answer :
Alright, let's address each part step-by-step by leveraging the principles of thermodynamics and the concept of specific heat capacity.
### (i) If the equal mass of [tex]\( X, Y \)[/tex], and [tex]\( Z \)[/tex] has the same temperature, which one has maximum heat?
The heat content [tex]\( Q \)[/tex] of a substance is given by the formula:
[tex]\[ Q = mc\Delta T \][/tex]
where [tex]\( m \)[/tex] is the mass, [tex]\( c \)[/tex] is the specific heat capacity, and [tex]\( \Delta T \)[/tex] is the change in temperature. Since we are given equal masses and the same temperature change for materials [tex]\( X, Y \)[/tex], and [tex]\( Z \)[/tex], the heat content will directly depend on their specific heat capacities.
Given specific heats are:
- [tex]\( c_X \)[/tex] = 420 J/(kg·°C)
- [tex]\( c_Y \)[/tex] = 380 J/(kg·°C)
- [tex]\( c_Z \)[/tex] = 470 J/(kg·°C)
Material [tex]\( Z \)[/tex] has the highest specific heat capacity (470 J/(kg·°C)), so [tex]\( Z \)[/tex] will have the maximum heat.
### (ii) If three pieces of them have equal temperature and equal amount of heat, which one of them has maximum mass?
For this scenario, since the heat content [tex]\( Q \)[/tex] and temperature change [tex]\( \Delta T \)[/tex] are the same for each material, we need to find which material would require the most mass under these conditions. Rearranging the heat formula to solve for mass [tex]\( m \)[/tex]:
[tex]\[ m = \frac{Q}{c\Delta T} \][/tex]
Given that [tex]\( Q \)[/tex] and [tex]\( \Delta T \)[/tex] are constant for all:
- For [tex]\( X \)[/tex]: [tex]\( m_X = \frac{Q}{420 \Delta T} \)[/tex]
- For [tex]\( Y \)[/tex]: [tex]\( m_Y = \frac{Q}{380 \Delta T} \)[/tex]
- For [tex]\( Z \)[/tex]: [tex]\( m_Z = \frac{Q}{470 \Delta T} \)[/tex]
The material with the lowest specific heat capacity requires the most mass to achieve the same amount of heat. Therefore, material [tex]\( Y \)[/tex] has the maximum mass because it has the lowest specific heat capacity (380 J/(kg·°C)).
### (iii) What do you mean by specific heat capacity of ' [tex]\( Z \)[/tex] ' is 470 J / (kg·°C)?
This statement means that for material [tex]\( Z \)[/tex], it takes 470 joules of energy to raise the temperature of 1 kilogram of [tex]\( Z \)[/tex] by 1 degree Celsius. It quantifies the amount of heat energy required to change the temperature of the substance.
### (iv) If the equal mass having the same shape and size of them at [tex]\( 100^{\circ} C \)[/tex] temperature is kept over a wax slab, which of them will melt the wax for maximum depth?
When equal masses of [tex]\( X, Y \)[/tex], and [tex]\( Z \)[/tex] at the same temperature are placed on a wax slab, the material with the highest heat content will transfer more heat to the wax, causing it to melt deeper. As established earlier, the material with the highest specific heat capacity will hold more heat. Therefore:
- [tex]\( c_X \)[/tex] = 420 J/(kg·°C)
- [tex]\( c_Y \)[/tex] = 380 J/(kg·°C)
- [tex]\( c_Z \)[/tex] = 470 J/(kg·°C)
Material [tex]\( Z \)[/tex] has the highest specific heat capacity and thus will melt the wax for the maximum depth.
In summary:
- i. Material [tex]\( Z \)[/tex] has the maximum heat.
- ii. Material [tex]\( Y \)[/tex] has the maximum mass.
- iii. The specific heat capacity of [tex]\( Z \)[/tex] means that it requires 470 joules to raise 1 kilogram of [tex]\( Z \)[/tex] by 1 degree Celsius.
- iv. Material [tex]\( Z \)[/tex] will melt the wax for maximum depth.
### (i) If the equal mass of [tex]\( X, Y \)[/tex], and [tex]\( Z \)[/tex] has the same temperature, which one has maximum heat?
The heat content [tex]\( Q \)[/tex] of a substance is given by the formula:
[tex]\[ Q = mc\Delta T \][/tex]
where [tex]\( m \)[/tex] is the mass, [tex]\( c \)[/tex] is the specific heat capacity, and [tex]\( \Delta T \)[/tex] is the change in temperature. Since we are given equal masses and the same temperature change for materials [tex]\( X, Y \)[/tex], and [tex]\( Z \)[/tex], the heat content will directly depend on their specific heat capacities.
Given specific heats are:
- [tex]\( c_X \)[/tex] = 420 J/(kg·°C)
- [tex]\( c_Y \)[/tex] = 380 J/(kg·°C)
- [tex]\( c_Z \)[/tex] = 470 J/(kg·°C)
Material [tex]\( Z \)[/tex] has the highest specific heat capacity (470 J/(kg·°C)), so [tex]\( Z \)[/tex] will have the maximum heat.
### (ii) If three pieces of them have equal temperature and equal amount of heat, which one of them has maximum mass?
For this scenario, since the heat content [tex]\( Q \)[/tex] and temperature change [tex]\( \Delta T \)[/tex] are the same for each material, we need to find which material would require the most mass under these conditions. Rearranging the heat formula to solve for mass [tex]\( m \)[/tex]:
[tex]\[ m = \frac{Q}{c\Delta T} \][/tex]
Given that [tex]\( Q \)[/tex] and [tex]\( \Delta T \)[/tex] are constant for all:
- For [tex]\( X \)[/tex]: [tex]\( m_X = \frac{Q}{420 \Delta T} \)[/tex]
- For [tex]\( Y \)[/tex]: [tex]\( m_Y = \frac{Q}{380 \Delta T} \)[/tex]
- For [tex]\( Z \)[/tex]: [tex]\( m_Z = \frac{Q}{470 \Delta T} \)[/tex]
The material with the lowest specific heat capacity requires the most mass to achieve the same amount of heat. Therefore, material [tex]\( Y \)[/tex] has the maximum mass because it has the lowest specific heat capacity (380 J/(kg·°C)).
### (iii) What do you mean by specific heat capacity of ' [tex]\( Z \)[/tex] ' is 470 J / (kg·°C)?
This statement means that for material [tex]\( Z \)[/tex], it takes 470 joules of energy to raise the temperature of 1 kilogram of [tex]\( Z \)[/tex] by 1 degree Celsius. It quantifies the amount of heat energy required to change the temperature of the substance.
### (iv) If the equal mass having the same shape and size of them at [tex]\( 100^{\circ} C \)[/tex] temperature is kept over a wax slab, which of them will melt the wax for maximum depth?
When equal masses of [tex]\( X, Y \)[/tex], and [tex]\( Z \)[/tex] at the same temperature are placed on a wax slab, the material with the highest heat content will transfer more heat to the wax, causing it to melt deeper. As established earlier, the material with the highest specific heat capacity will hold more heat. Therefore:
- [tex]\( c_X \)[/tex] = 420 J/(kg·°C)
- [tex]\( c_Y \)[/tex] = 380 J/(kg·°C)
- [tex]\( c_Z \)[/tex] = 470 J/(kg·°C)
Material [tex]\( Z \)[/tex] has the highest specific heat capacity and thus will melt the wax for the maximum depth.
In summary:
- i. Material [tex]\( Z \)[/tex] has the maximum heat.
- ii. Material [tex]\( Y \)[/tex] has the maximum mass.
- iii. The specific heat capacity of [tex]\( Z \)[/tex] means that it requires 470 joules to raise 1 kilogram of [tex]\( Z \)[/tex] by 1 degree Celsius.
- iv. Material [tex]\( Z \)[/tex] will melt the wax for maximum depth.
